This could be a really powerful feature for sidechains - the benefits of merge mining but without needing Bitcoin miners to do anything.
But if I understood it right, someone or a group needs to keep paying the Bitcoin to keep it going. Who would pay if the sidechain is decentralised? (like with proof of work or proof of stake)
If the sidechain is federated, then the block-signing federation is responsible for funding the Bitcoin staychain - and the Bitcoin staychain can be run with the same multisig script for signing sidechain blocks (we show in the paper how you can verify a pay-to-contract homomorphic commitment via a multisig redeem script).
For a fully decentralised/permissionless sidechain, it would be possible to incorporate SPV proofs of Bitcoin staychain transactions into the block reward policy of the sidechain: in this way sidechain miners would be incentivised to fund the Bitcoin staychain in order to gain block rewards on the sidechain. The details of this would need to be worked out though.