feat: Add support for Conversations in Responses API (#3743)

# What does this PR do?
This PR adds support for Conversations in Responses.

<!-- If resolving an issue, uncomment and update the line below -->
<!-- Closes #[issue-number] -->

## Test Plan
Unit tests
Integration tests

<Details>
<Summary>Manual testing with this script: (click to expand)</Summary>

```python
from openai import OpenAI

client = OpenAI()
client = OpenAI(base_url="http://localhost:8321/v1/", api_key="none")

def test_conversation_create():
    print("Testing conversation create...")
    conversation = client.conversations.create(
        metadata={"topic": "demo"},
        items=[
            {"type": "message", "role": "user", "content": "Hello!"}
        ]
    )
    print(f"Created: {conversation}")
    return conversation

def test_conversation_retrieve(conv_id):
    print(f"Testing conversation retrieve for {conv_id}...")
    retrieved = client.conversations.retrieve(conv_id)
    print(f"Retrieved: {retrieved}")
    return retrieved

def test_conversation_update(conv_id):
    print(f"Testing conversation update for {conv_id}...")
    updated = client.conversations.update(
        conv_id,
        metadata={"topic": "project-x"}
    )
    print(f"Updated: {updated}")
    return updated

def test_conversation_delete(conv_id):
    print(f"Testing conversation delete for {conv_id}...")
    deleted = client.conversations.delete(conv_id)
    print(f"Deleted: {deleted}")
    return deleted

def test_conversation_items_create(conv_id):
    print(f"Testing conversation items create for {conv_id}...")
    items = client.conversations.items.create(
        conv_id,
        items=[
            {
                "type": "message",
                "role": "user",
                "content": [{"type": "input_text", "text": "Hello!"}]
            },
            {
                "type": "message",
                "role": "user",
                "content": [{"type": "input_text", "text": "How are you?"}]
            }
        ]
    )
    print(f"Items created: {items}")
    return items

def test_conversation_items_list(conv_id):
    print(f"Testing conversation items list for {conv_id}...")
    items = client.conversations.items.list(conv_id, limit=10)
    print(f"Items list: {items}")
    return items

def test_conversation_item_retrieve(conv_id, item_id):
    print(f"Testing conversation item retrieve for {conv_id}/{item_id}...")
    item = client.conversations.items.retrieve(conversation_id=conv_id, item_id=item_id)
    print(f"Item retrieved: {item}")
    return item

def test_conversation_item_delete(conv_id, item_id):
    print(f"Testing conversation item delete for {conv_id}/{item_id}...")
    deleted = client.conversations.items.delete(conversation_id=conv_id, item_id=item_id)
    print(f"Item deleted: {deleted}")
    return deleted

def test_conversation_responses_create():
    print("\nTesting conversation create for a responses example...")
    conversation = client.conversations.create()
    print(f"Created: {conversation}")

    response = client.responses.create(
      model="gpt-4.1",
      input=[{"role": "user", "content": "What are the 5 Ds of dodgeball?"}],
      conversation=conversation.id,
    )
    print(f"Created response: {response} for conversation {conversation.id}")

    return response, conversation

def test_conversations_responses_create_followup(
        conversation,
        content="Repeat what you just said but add 'this is my second time saying this'",
    ):
    print(f"Using: {conversation.id}")

    response = client.responses.create(
      model="gpt-4.1",
      input=[{"role": "user", "content": content}],
      conversation=conversation.id,
    )
    print(f"Created response: {response} for conversation {conversation.id}")

    conv_items = client.conversations.items.list(conversation.id)
    print(f"\nRetrieving list of items for conversation {conversation.id}:")
    print(conv_items.model_dump_json(indent=2))

def test_response_with_fake_conv_id():
    fake_conv_id = "conv_zzzzzzzzz5dc81908289d62779d2ac510a2b0b602ef00a44"
    print(f"Using {fake_conv_id}")
    try:
        response = client.responses.create(
          model="gpt-4.1",
          input=[{"role": "user", "content": "say hello"}],
          conversation=fake_conv_id,
        )
        print(f"Created response: {response} for conversation {fake_conv_id}")
    except Exception as e:
        print(f"failed to create response for conversation {fake_conv_id} with error {e}")


def main():
    print("Testing OpenAI Conversations API...")

    # Create conversation
    conversation = test_conversation_create()
    conv_id = conversation.id

    # Retrieve conversation
    test_conversation_retrieve(conv_id)

    # Update conversation
    test_conversation_update(conv_id)

    # Create items
    items = test_conversation_items_create(conv_id)

    # List items
    items_list = test_conversation_items_list(conv_id)

    # Retrieve specific item
    if items_list.data:
        item_id = items_list.data[0].id
        test_conversation_item_retrieve(conv_id, item_id)

        # Delete item
        test_conversation_item_delete(conv_id, item_id)

    # Delete conversation
    test_conversation_delete(conv_id)

    response, conversation2 = test_conversation_responses_create()
    print('\ntesting reseponse retrieval')
    test_conversation_retrieve(conversation2.id)

    print('\ntesting responses follow up')
    test_conversations_responses_create_followup(conversation2)

    print('\ntesting responses follow up x2!')

    test_conversations_responses_create_followup(
        conversation2,
        content="Repeat what you just said but add 'this is my third time saying this'",
    )

    test_response_with_fake_conv_id()

    print("All tests completed!")


if __name__ == "__main__":
    main()
```
</Details>

---------

Signed-off-by: Francisco Javier Arceo <farceo@redhat.com>
Co-authored-by: Ashwin Bharambe <ashwin.bharambe@gmail.com>

docs/static/deprecated-llama-stack-spec.html

@@ -10083,6 +10083,10 @@
                         "type": "string",
                         "description": "(Optional) if specified, the new response will be a continuation of the previous response. This can be used to easily fork-off new responses from existing responses."
                     },
+                    "conversation": {
+                        "type": "string",
+                        "description": "(Optional) The ID of a conversation to add the response to. Must begin with 'conv_'. Input and output messages will be automatically added to the conversation."
+                    },
                     "store": {
                         "type": "boolean"
                     },

docs/static/deprecated-llama-stack-spec.yaml

@@ -7493,6 +7493,12 @@ components:
             (Optional) if specified, the new response will be a continuation of the
             previous response. This can be used to easily fork-off new responses from
             existing responses.
+        conversation:
+          type: string
+          description: >-
+            (Optional) The ID of a conversation to add the response to. Must begin
+            with 'conv_'. Input and output messages will be automatically added to
+            the conversation.
         store:
           type: boolean
         stream:

docs/static/llama-stack-spec.html

@@ -8178,6 +8178,10 @@
                         "type": "string",
                         "description": "(Optional) if specified, the new response will be a continuation of the previous response. This can be used to easily fork-off new responses from existing responses."
                     },
+                    "conversation": {
+                        "type": "string",
+                        "description": "(Optional) The ID of a conversation to add the response to. Must begin with 'conv_'. Input and output messages will be automatically added to the conversation."
+                    },
                     "store": {
                         "type": "boolean"
                     },

docs/static/llama-stack-spec.yaml

@@ -6189,6 +6189,12 @@ components:
             (Optional) if specified, the new response will be a continuation of the
             previous response. This can be used to easily fork-off new responses from
             existing responses.
+        conversation:
+          type: string
+          description: >-
+            (Optional) The ID of a conversation to add the response to. Must begin
+            with 'conv_'. Input and output messages will be automatically added to
+            the conversation.
         store:
           type: boolean
         stream:

docs/static/stainless-llama-stack-spec.html

@@ -10187,6 +10187,10 @@
                         "type": "string",
                         "description": "(Optional) if specified, the new response will be a continuation of the previous response. This can be used to easily fork-off new responses from existing responses."
                     },
+                    "conversation": {
+                        "type": "string",
+                        "description": "(Optional) The ID of a conversation to add the response to. Must begin with 'conv_'. Input and output messages will be automatically added to the conversation."
+                    },
                     "store": {
                         "type": "boolean"
                     },

docs/static/stainless-llama-stack-spec.yaml

@@ -7634,6 +7634,12 @@ components:
             (Optional) if specified, the new response will be a continuation of the
             previous response. This can be used to easily fork-off new responses from
             existing responses.
+        conversation:
+          type: string
+          description: >-
+            (Optional) The ID of a conversation to add the response to. Must begin
+            with 'conv_'. Input and output messages will be automatically added to
+            the conversation.
         store:
           type: boolean
         stream: