It's not so much about the originating realm as it is about the realm of the noble in prison.
You can execute advies, rogues, and those banished from your realm. So, if the realm he is from banishes him while he is in prison and he goes rogue, they can execute him regardless of whether he is banished from his captor's realm. This is most often (though still quite rarely) seen when a noble is captured by their own realm for whatever reason and the judge holds them in prison, banishes them from their realm, waits for them to go rogue (perhaps torturing to prevent escape), and then executes.
Of course, in this instance, the banished noble could decide to join his captor's realm instead of going rogue. And then they'd have to banish them. Which would extend time before execution or escape (and he's only held 7 days anyway).
