If not on Lightning it will come to other L2s on Bitcoin. Especially Bitcon mainchain rollups with zero-knowledge proofing are in the pipeline. They want to be interoperable with as many other chains as possible. Cooperation with Monero as L2 on Bitcoin may then be possible. Exciting times ahead, as we all know 😉
Discussion
I don't think that such things will be widely used. Monero users use Monero directly for a reason. To use that on top of BTC is worthless. I think you talk about drivechains. The are not so cool as they sound. Fully custodial for example. I'm also not sure if the swap to another chain doesn't have a special footprint, if yes you can end up with tainted coins.
Too much to respond, but in short. No, I don't mean drivechains (I'm not familiar with it), but protocols using ZK-proofs built on BTC main chain, that can provide the security and decentralization of Bitcoin while being scalable for global use. This technology was first built on ETH and as far as I learned can now be built on BTC as well without requiring a fork. Projects which are at it currently are e.g. BitcoinOS and AlpenLabs. It would enable to build a (digital) economy and financial system on BTC. Without it BTC likely remains a stale asset. Lightning and Rootstock are cool, but lack scalability and compute potential so far. Have a look, it's interesting stuff! Monero this way could piggy bag off of the security and decentralization of Bitcoin and add its privacy feature, instead of struggling to be relevant as its own L1.