[Bug] Remote Control messages stall when status panel (QR code) is open in terminal

**Bug Description**
Title: Remote Control messages stall at "Preparing..." while /remote-control status panel (QR code) is open in the terminal

Description:
While the /remote-control status panel (session URL + QR code) is open in the local terminal, messages sent from claude.ai/code are not processed. The browser shows "Preparing..." indefinitely (observed 3+ minutes). Dismissing the panel immediately unblocks the session and the queued message is processed.

Steps to reproduce:
1. Start an interactive Claude Code session and run /rc.
2. Connect to the session from claude.ai/code in a browser.
3. In the terminal, run /rc again to open the status panel and show the QR code.
4. Leave the panel open. From the browser, send a short message (e.g. "I'm here").
5. Observe: browser shows "Preparing..." and nothing appears in the terminal.
6. Dismiss the panel in the terminal.
7. Observe: the message is processed immediately.

Expected:
Remote messages should be processed while the status panel is open, or the panel should not block the session. At minimum, the panel or docs should indicate that it pauses remote input.

Actual:
Remote input is silently blocked with no indication on either side until the panel is closed.

Environment:
Windows, Claude Code CLI (version auto-attached), claude.ai/code in browser.

**Environment Info**
- Platform: win32
- Terminal: windows-terminal
- Version: 2.1.273
- Feedback ID: 08f3d4e9-2201-424b-bb5f-52eeb5a1c655

**Errors**
```json
[]
```

Start by reproducing the issue through the /rc entry point in an interactive terminal session, with a connected claude.ai/code browser session. Trace what the terminal status panel does while it is open and compare that with remote message processing. Done means messages sent from the browser are processed while the QR-code panel is open, or the blocking behavior is clearly indicated.

Written by the indexing model from the issue text.