Ouch, my wife has encountered almost exactly that in a recent brush with a biotech company that seems to have been infected by FAANG expats. She was advised that any kind of sidebar conversation is a faux pas.
I struggled to guess at the real motive. Is it some project manager's blatant control freakery? An org-level, cynical management attempt to commoditize their knowledge workers? Or some kind of emergent failure where culture morphs through openness -> radical transparency -> enforced conformism a la 1984?
Probably cargo culting from companies with 50k+ employees (where minor friction for one is way outweighed by more availability for all).
It feels good as a manager to formalize things, but the best collaboration and ideas happen organically at less formal times and places - and those times are worth at least as much, if not more to the company than anything formal.
You might as well say "no thinking about work in the shower."