Is there any way for the public to verify that the bounty can actually be claimed by doing the work? Or do they have to rely on the honesty of the person announcing the bounty?

In other words, is there something that stops me from saying that a certain computation that I want people to do will allow claiming the bounty associated with a certain Bitcoin address, when in fact the private key is some completely distinct key that I control and that has nothing to do with the time-lock computation?