Chat input history leaks across sessions

## Type of issue

Bug

## Description

Chat input history navigation is shared across chat sessions. After submitting a prompt in one session, opening another session and pressing Up can recall the prompt from the previous session.

## Steps to reproduce

1. Open Chat and submit `SESSION_A_ONLY explain alpha`.
2. Start a new chat session.
3. Submit `SESSION_B_ONLY summarize beta`.
4. Focus the input and press Up more than once.

## Actual behavior

History navigation in Session B can recall the prompt submitted in Session A.


## Expected behavior

Input history is scoped to the active chat session. Session B should only recall prompts submitted in Session B, while returning to Session A should restore Session A history.

## Environment

Reproduced in a local Code - OSS web workbench at commit `60ef009ffa2c6c2bdc27cdf07e0656eafdd5bc9d` using a synthetic local chat participant.