Slitherine

Couldn't find a definitive answer in the manual, so here's my question:

What happens when Italy has already surrendered, and the Allies capture Hamburg and Berlin, but Rome and Paris are still Axis-controlled. My guess is: German surrender, Allied victory. Correct? Thanks for clarification.

If the Allies take Berlin and Hamburg then Germany surrenders, so German-controlled Paris and Rome also pass to the Allies

Tested it in Hotseat, and the answer is yes. I was sure of this but then got confused because the manual talks about the Axis having to lose their final capital. Just wanted to be sure, considering the potential consequences for a current game of mine...