Changing the chat mode picker (Agent → Ask/Plan) resends "Start implementation" and discards chat draft

### Summary
Changing the chat mode picker (Agent → Ask/Plan) while an unsent draft is in the
input box discards the draft and automatically resubmits the previous request.
In my session this repeatedly resubmitted `Start implementation`, which executed
in Agent mode and modified files without my intent.

### Steps to reproduce
1. Open Copilot Chat in Agent mode.
2. Send a request and let it complete.
3. Start typing a new prompt in the chat input, but do not send it.
4. Change the mode picker from Agent to Ask (or Plan).

### Expected
Changing mode preserves the unsent draft and submits nothing. Selecting a mode is
not a submit action.

### Actual
- The unsent draft text is discarded.
- The previous request is resubmitted automatically.
- No confirmation, and no chance to cancel before execution begins.

### Impact
This happened 4 times in a single session. Because the resubmitted request ran in
Agent mode, the agent performed substantial unrequested work: created and edited
files, deleted files, started local servers, and consumed premium requests.
Silent resubmission in Agent mode can cause real and sometimes irreversible
workspace changes.

### Additional detail not yet covered in this issue
- Reproduces on **Agent → Ask**, so it is not limited to Plan mode.
- **The in-progress draft is lost**, which I did not see reported above.
- Triggered by the **mode picker itself**, not only the back/return button
described in #324436.

### Related
- #324436 — closed as duplicate; reports the identical literal string
`Start implementation`
- #323992 — closed, milestone 1.126.0; behavior still reproduces for me

### Environment
Version: 1.134.0 (user setup)
Commit: 110a328ea54b42367b803ec53ee0bf52ef26b419
Date: 2026-08-18T18:24:44Z
Electron: 42.8.1
ElectronBuildId: 14906494
Chromium: 148.0.7778.280
Node.js: 24.18.1
V8: 14.8.178.38-electron.0
@github/copilot: 1.0.81-0
@github/copilot-sdk: 1.0.11
OS: 110a328ea54b42367b803ec53ee0bf52ef26b419