side-question (/btw) endpoint is a stub returning hardcoded placeholder; implement real handler

## Summary

The `/btw` "side question" (by-the-way) feature in AionUi does not work: when a user runs `/btw ` inside a Claude Code (acp/claude) conversation, the SIDE QUESTION overlay shows a hardcoded placeholder answer instead of a real answer:

> Side question support will be fully wired in app integration phase.

## Root cause (confirmed from the AionUi side)

The AionUi front-end is **fully wired** — the `/btw` command parsing, the overlay UI, the IPC bridge, the endpoint call, and the response handling all work correctly. The placeholder string does **not** exist anywhere in the AionUi TypeScript/JS source, i18n, mocks, or fixtures — it is only present inside the compiled `aioncore` backend binary.

The endpoint is:
```
POST /api/conversations/{conversation_id}/side-question
body: { "question": "..." }
```
Response shape expected by the client:
```
{ status: 'ok'; answer: string }
| { status: 'noAnswer' }
| { status: 'unsupported' }
| { status: 'invalid'; reason: 'emptyQuestion' }
| { status: 'toolsRequired' }
```

Currently the backend handler for this endpoint appears to be a **stub** that returns `{ status: 'ok', answer: "Side question support will be fully wired in app integration phase." }` — i.e. it always answers with the placeholder text, so the feature looks broken/"integration failed" to users.

## Expected

Implement the real side-question handler in `aioncore` so `/btw` actually asks the side question against the running Claude Code conversation and returns a genuine `answer` (or the appropriate `noAnswer` / `unsupported` / `toolsRequired` status), instead of the hardcoded placeholder.

## Notes

- Not fixable from the AionUi (`packages/desktop`) repo — the TS layer already forwards the request and renders whatever `answer` the backend returns. The fix must land in this backend crate and ship in a rebuilt `aioncore` binary.
- Front-end eligibility gate: `/btw` is only offered when `type === 'acp' && backend === 'claude'`.

Start at the aioncore backend handler for POST /api/conversations/{conversation_id}/side-question and trace how it accesses the running acp/claude conversation. Replace the hardcoded placeholder behavior with the real side-question flow, returning a genuine answer or the documented noAnswer, unsupported, invalid, or toolsRequired status.

Written by the indexing model from the issue text.