Using /btw opens a split chat instead of the subchat treatment

### Description

When using `/btw` with the Copilot harness, VS Code opens a second chat side by side and reuses the loading indicator from the original chat. This is confusing because the new conversation appears related to the original chat, but is presented as a separate split chat.

### Steps to Reproduce

1. Open a chat using the Copilot harness.
2. Run `/btw` while the chat is active.
3. Observe that a second chat opens beside the original chat and the original chat's loading indicator is reused.

### Expected Behavior

Present the `/btw` conversation using the same visual treatment as a subchat: tabs at the top and an indented child entry in the sessions list. If `/btw` is intentionally a separate chat at the SDK level, the UI should still make that relationship and loading state clear.

### Actual Behavior

Two chats are shown side by side, and the loading indicator appears on the original chat, making it unclear which conversation is active or processing.