New POST heaviest-chain finder without downloading any blocks or trusting a checkpoint?

In a POST consensus mechanism with no starting stake needed, I may have found a way to sync with the correct chain without a.) trusting checkpoinnts or b.) downloading the whole chain (or even part of it). Simply, make a tiny POST proof, hashes of you address plus a nonce at the end. Once you find a successful POST proof, upload it to the blockchain as a "sync fee." You can do this (and should do this) for all competing chains. The fee is very tiny since you can reserve extra block space for it, and because you can't use the fee for anything other than syncing, and the syncing is paid for by the fee, so you don't even need a signature. You can do it with computational ease. Anyways, you pay a tiny, tiny sync fee kon all competing chains. Then, say, 6 blocks into the future (to prevent tampering) it affects the randomness beacon (which uses something like a VRF, not RanDAO.) Any attacking chain that mines blocks ahead of time cannot accept the transaction, or their blocks would be invalid; however, the correct chain would likely accept, allowing you to disinguish between valid and invalid. Would this work? Feedback?