In many states, once the apartment is relet the lease to the previous tenant is terminated.
With the current housing situation I can't see a landlord having troubles getting a new tenant in there immediately. Of course there is incentive to keep it empty and keep the lessee paying the bill so it's best if he talks to the landlord to feel out what they would do. Of course, if it's an "all utilities paid" place, those costs are wrapped into the lease payment so there is even more incentive for them to keep it empty and turn the heat off and keep collecting the lease payment.
ETA, if that were my son he is on the hook for the payments. If he doesn't pay and I'm forced to pay to protect my credit, I'm amending my trust to read, "Johnny gets his share of the assets minus $XYZ and compounded interest at the rate of 10% from January 2023."