Desktop app Web tab shows 'No web sessions yet' despite sessions existing on claude.ai/code

Bug Description

The desktop app's session picker "Web" tab shows "No web sessions yet", but sessions clearly exist when accessing claude.ai/code in a browser with the same account.

Steps to Reproduce

1. Create sessions on claude.ai/code (web browser)
2. Open Claude Code desktop app (logged in with the same account)
3. Click the session picker and switch to the "Web" tab
4. Web tab shows "No web sessions yet" with a link to "Start one at claude.ai/code"

Expected Behavior

Web sessions created on claude.ai/code should appear in the desktop app's Web tab.

Actual Behavior

Web tab is empty despite multiple sessions existing on claude.ai/code.

Screenshots

Desktop app Web tab (empty):
The Web tab in the session picker shows "No web sessions yet".

claude.ai/code in browser (has sessions):
Multiple sessions visible in the left sidebar (e.g., "New session", "Connect to server", "Review and investigate open issues", etc.)

Environment

• Platform: macOS (Darwin 25.3.0)
• Same account used on both desktop app and web