[bug] "✗ Auto-update failed · Run /doctor" in status bar causes background agents to stop during long-running sessions

Summary

During long-running background agent sessions (babysit / agents mode), the status bar shows "✗ Auto-update failed · Run /doctor" and the agent stops or hangs.

Steps to reproduce

1. Start a long-running background Claude agent session (agents mode / babysit mode)
2. Leave it running for an extended period
3. Observe: "✗ Auto-update failed · Run /doctor" appears in the status bar footer
4. The agent stops or hangs — session does not recover automatically

Expected behavior

Auto-update checks should either:

• Not run during active agent sessions, or
• Fail gracefully without disrupting the running agent

Actual behavior

The auto-update check fires mid-session and fails. The failure surfaces in the status bar as "✗ Auto-update failed · Run /doctor" and causes the agent to stop/hang rather than continuing with the current version.

Environment

• Claude Code version: 2.1.161
• OS: macOS (darwin)
• Mode: background agents / babysit mode

Additional notes

This is a recurring issue across multiple long-running sessions, not a one-off occurrence. The update check appears to be scheduled on a timer that fires regardless of whether an active agent session is in progress.