Ownership of keys doesn't need to be proved unless you are signing a message or broadcasting a transaction. Moreover, for most bitcoin hodlers it is considered absolutely normal to never disclose any ownership over specific UTXOs for privacy reasons. Not your keys, not your bitcoins. My keys, not your business, sorry. I don't have to prove anything as long as I keep my private keys secret. I don't have to prove that I am hiding something.

Proof of Stupidity (PoS) is a more appropriate name for this movement. They are trying to prove something while paying high fees to exchanges to withdraw "their" coins. And then, they prove they can't control the keys for long and deposit bitcoins back. I wonder why rational bitcoiners ignore that movement.