[BUG] Chat tab strip (multi-session view) missing in VS Code extension after update to 2.1.273

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

After updating the Claude Code VS Code extension to 2.1.273 (VS Code 1.138.0, macOS), the tab strip that used to appear at the top of the Claude Code panel is gone. This tab strip used to show all active/recent chats so I could switch between them without losing my place. Previously, clicking the "+" (new chat) or clock (history) icon would open additional chats as tabs across the top of the panel. Now it only switches/replaces the current session, and there is no way to see multiple open chats at once inside the Claude Code extension panel.

Troubleshooting already tried, none resolved it:
- Fully quit and reopened VS Code
- Restarted the computer
- Confirmed extension is on latest version (2.1.273)
- Confirmed VS Code is on latest version (1.138.0)

Remote Control also briefly disconnected around the same update, but that was separately resolved by fully quitting and reopening VS Code.

### What Should Happen?

Chats should open as tabs in the Claude Code panel header, so I can switch between multiple active conversations without closing or losing any of them, same as before updating to 2.1.273.

### Error Messages/Logs

```shell

```

### Steps to Reproduce

1. Open the Claude Code panel in VS Code (previously had "Chat" and "Claude Code" tabs at top, with a tab strip showing active chats).
2. Update the Claude Code extension to 2.1.273 and/or VS Code to 1.138.0.
3. Click the "+" (new chat) or clock (history) icon to open another chat.
4. Observe: the new/selected chat replaces the current view instead of opening as an additional tab in a tab strip.

### Claude Model

None

### Is this a regression?

Yes, this worked in a previous version

### Last Working Version

Version before 2.1.273 (exact last-working version unknown)

### Claude Code Version

2.1.273 (VS Code extension)

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

Related closed/locked issue with a similar root cause on the VS Code extension: #64515 (closed as "not planned" for inactivity, then auto-locked). Also possibly related: #55181 ("Claude code tab disappeared after version upgrade").

Also noticed: Remote Control keeps disconnecting/needing to be reconnected around the same update, where before this update it stayed connected persistently across restarts without me needing to do anything.

Start by reproducing the missing tab strip in the Claude Code VS Code panel on extension 2.1.273, using the + and clock controls to open multiple chats. Compare the behavior with a version before 2.1.273; done means active and recent chats appear as separate tabs and switching between them does not replace or lose the current session.

Written by the indexing model from the issue text.