Immobilizer issue??

[Rob175]

My 1998 base Boxster has 2 key fobs each with 2 buttons (one for the door locks and one for the rear trunk). They both worked OK but just OK....meaning that I always had to be standing very close (within 2 to 3 ft) to the car for the buttons to do their thing.

Seemingly suddenly BOTH key fobs no longer lock/unlock the doors and no longer open the rear trunk. One day they worked and the next day they don't anymore.

Both have brand new Duracell batteries (tried several new batteries in each). Both fobs little red lights blink properly when pressed but nothing else.

The keys work fine in the ignition and the car starts perfectly......

Am I correct in thinking the fob's buttons send a signal to the "immobilizer" or is there some other system that controls the remote locking/unlocking? Is there an antenna of sorts that receives the signals from the fob's button pressing?

It's not a terrible thing to turn the key in the slot to lock/unlock the doors but I'd like it to work as it previously did without spending a fortune of taking apart the whole car...LOL

Any insight would help...thx