[FEATURE] keep a log of /btw messages the user asks

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

In academic writing we are often required to submit a protocol of genAI interactions. The /btw chats apparently do not get logged, and also do not appear in the terminal buffer. So I cannot submit a full protocol (I would even acknowledge that the /btw model is not a strong one). This might be related to issues 79593, 81736, and 83648

Proposed Solution

Write a log of /btw messages and responses to disk

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

Developer tools/SDK

Use Case Example

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Additional Context

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