That’s only valid under the assumption that there is no collusion with the next validator. This may not hold in practice. Validators can publish their preimages if they wish and everyone can verify them.
To avoid coordination (collusion), we could set up a early reveal penalization scheme such that everyone knowing H^{-1}(C_{V_i}) of a validator can publish it to have the deposit of the validator steal a part of the validator deposit and have another part burnt.
This way, if some validator were to publish its preimage early, everyone could steal ETH from it.
Getting a transaction showing an early preimage reveal included in a block, may in practice, only be doable by the previous validator (because it would prefer to make this transaction itself than including the transaction from another party). But that still prevent trustless reveal of one’s preimage to the previous validator.
But, this preimage penalization scheme could be bypassed by validators colluding. Colluding validators could make a smart contract which would penalized them (by burning part of ETH staked in a smart contract), if they include an “early reveal penalization” transaction in a block they make, thus parties wanting to collude with this validator would know that they can safely reveal it their preimage as soon as it is its turn to make a block.
Again, if there is collusion, multiple validators could skip their turn in a row if this makes them being drawn more.