Yes, it's the same kind of Proof of Stake, but it has a chain. And that difference allows direct payments between participants, and that's the key difference between my proposal and Lightning Network: it is like if you could watch all channels and their state. Also, in this case, no watchtower is ever needed, and mining is also possible, because if you can watch all channels and transactions, then you can collect all of them, make a sidechain block out of it, and sign it, then you take all fees from all users as your reward (by default your reward is zero, and can be increased by validating transactions). So, the situation is clear: people can wait a long time for mainchain confirmation and pay on-chain fees, but they can instead use minimal on-chain fees, and pay much less, to have it signed by stakers. In this way, the stakechain can fill the gap between zero and the first confirmation, for a few satoshis you can get it validated, instead of accepting zero-confirmation. No coins will be created out of thin air, users will agree voluntarily to move their coins into stakers' hands, just by signing transactions.

You can, I guess that will be the case during bootstrapping, then people will find a good balance between that, and will see that staking can give them more profits, while having less power requirements, because then no mining equipment is needed. It will be a gradual way from 100% mining and 0% staking into 1% mining (the minimal difficulty is one) and 99% staking.

It has to be gradual, because instant change will be rejected by Proof of Work enthusiasts. They don't want to switch immediately, they have to be convinced over time, that a different system is better, and can reward them with more coins.