Desktop: Chat/Cowork mode picker only appears on cold app launch

What happens

The Chat/Cowork mode toggle is visible on the empty composer when the desktop app is launched. Once a conversation has started, the toggle disappears and that conversation is permanently locked to the mode it began in.

Cmd+N (new conversation) returns to the picker, but there is no way to switch an in-flight conversation.

Expected

Mode should be switchable mid-conversation, or the picker should remain reachable without starting a new conversation.

Impact

Choosing the wrong mode means abandoning the thread and re-establishing context in a fresh one.

Environment

Claude desktop app, macOS.

Related

#50341 — with Cmd+1/Cmd+2/Cmd+3 not working, a hidden picker leaves no switching path at all.