Expose Remote Control connection status in the statusLine JSON payload

**Feature request**

Custom `statusLine` scripts (configured via `settings.json` → `statusLine.command`) receive a JSON payload on stdin, documented at [statusline.md](https://code.claude.com/docs/en/statusline). The documented fields (`model`, `workspace`, `context_window`, `rate_limits`, `pr`, `vim.mode`, `agent.name`, etc.) do not include anything about Remote Control connection state.

Meanwhile, [remote-control.md](https://code.claude.com/docs/en/remote-control#check-connection-status) describes a native `/rc active` indicator that shows in the footer (or startup header, in fullscreen rendering) while a Remote Control connection is up — but this indicator is entirely separate from the custom statusline mechanism, and disappears from view once the conversation scrolls past the startup header (in fullscreen mode).

**Problem:** anyone running a custom statusline script has no way to reflect Remote Control's live connection status in it, since the data simply isn't passed to the script. The only options today are:
- a static/always-on hint in the custom statusline (misleading — doesn't reflect actual state), or
- rely on the native indicator, which is only reliably visible at session start in fullscreen mode.

**Proposed fix:** add a `remote_control` object to the statusline JSON payload, e.g.:

```json
"remote_control": {
"connected": true,
"url": "https://claude.ai/code/..."
}
```

This would let custom statusline scripts render an accurate, persistent, live RC status indicator alongside git branch, context usage, etc. — consistent with how `pr.number`/`pr.url` already mirror the footer's PR badge.

**Version:** Claude Code 2.1.270

Start by reading statusline.md and the Remote Control connection-status section in remote-control.md to compare the documented payload with the existing indicator. Trace where the statusLine JSON is assembled and where live Remote Control state is available; done means custom statusline scripts receive the proposed remote_control fields with accurate connected state and URL.

Written by the indexing model from the issue text.