They COULD do it, just seems doubtful they WOULD do it.
As mentioned, there is already a loophole you can exploit. And the reason the loophole is there is because we have separate systems in handling active and reserve processes. Also, as mentioned, you don't have to sign a thing and you will still promote once your name pops up on the NAVADMIN. The paperwork is truly just a formality - don't even recall seeing it in my OMPF (my origianal oath is, but not my promotion paperwork).
The way it's setup now, there is nothing to stop someone from signing an agreement, pinning rank and walking right over to reserve at same rank before that time is up. The current time in grade needed for O4 is 6 months (another element to address). You would have to rely on the reserve system to catch the person and demote them to previous rank. We already only have minimal folks working reserve processes (including retirement, the motivation behind this thread). And said paperwork would need to transfer from active to reserve to even have a chance of catching it!
So if reserve side isn't staffed to catch it when affiliation occurs, it would fall on the active side to catch it before you depart. The same process that already handles most things electronically (DD214, etc). You don't face to face with anyone when separating beyond TGPS/TAPS.
Who acts as gatekeeper for this system? The answer would be to bridge all processes active and reserve into one system - but this hardly seems the catalyst to make that happen. It's not THAT serious.