Check if you're in combat. You can interact with the calendar while in combat, but can't touch the release buttons while there are enemies after you in previous rooms (and you can't go back anymore for whatever the reason).

Because the cages activate one by one after a short while without any interactions (and thus spawning new enemies), you're not actually engaged in combat with enemies long enough for them to back off, resetting the timer and being forced to just stand there and wait for the entire situation to slowly play itself out.

In conclusion - kill what you pull.