Signal Desktop doesn't require the phone, so no, it doesn't work that way.
Suppose I install and configure Signal/Phone, then install and configure Signal/Desktop, then turn off the phone.
Will Signal/Desktop still work while the phone is turned off?
* If yes, then what you say appears to be wrong.
* If no, then that's genuinely astonishing to me, and I've learning something.
Is there a grace period ?