Sidebar groups don't sync across devices (desktop only, absent in mobile Code list)

### What happens

Custom sidebar groups in the Code tab exist only in the desktop app. Signed in to the same account in the Claude mobile app, the Code session list shows the sessions themselves — including remote-control (`/rc`) sessions running on another machine — but no groups, and no way to see or use the grouping.

### Expected

Sidebar groups sync with the account, the same way session titles and archive state already do.

### Why this looks like a layering gap rather than a hard problem

Session titles already cross the device boundary and work fine as identifiers on mobile. Groups do not.

Locally, a session's stored record (`~/Library/Application Support/Claude/claude-code-sessions///.json`) contains `title`, `cwd`, `model`, `isArchived`, `permissionMode`, `lastActivityAt`, `sessionId` — but no group field at all. The group appears to be desktop sidebar view state reported by the app window at runtime, rather than part of the synced session data. So the sync path itself clearly exists; grouping just isn't on it.

### Impact

`/rc` sessions are precisely the ones viewed from more than one device — that is the point of remote control. A phone list is also narrower, so titles truncate and grouping would help there most of all.

Today the only cross-device organizing mechanism is encoding the host into the session title by hand, e.g. `[CT110] project - task`. That works, but it means maintaining a naming convention to substitute for a feature that already exists on one surface.

### Environment

- Claude Desktop on macOS, Code tab
- Claude mobile app
- Remote-control sessions started with `/rc` in headless Claude Code on a Linux container
- Same account on all surfaces

Start by tracing the session record at ~/Library/Application Support/Claude/claude-code-sessions///.json and the existing account sync for titles and archive state. Compare how desktop Code sidebar groups and mobile Code sessions are represented, including /rc sessions. Done means groups synchronize and are visible and usable on both surfaces without regressing existing session sync.

Written by the indexing model from the issue text.