In this case there would be no trust involved, the delegation would be a result of a cryptographic process/proof of work challenge. Ignore any reliance on "trust".
I'm wondering if such mechanism is possible and how.

first and second rounds might not be something relevant, that's just how I would put it in english. If there's a delagated proof of work type of consensus, then there is two "steps" with two "methods", that's what I meant.

What I'm wondering is if there's a way to prevent the situation explained earlier in this thread, nodes switching pool in order to bypass the first step or round, which would be some kind of cheating. It doesn't have to be knowing the hashpower, in fact I have no idea what it would be, just interested to know if what I've explained is possible and why.

Yes I guess that would purely be about the use of electricity.

Is there a way to programmatically detect if the hashpower (or something related) of the nodes/pool in an hypothetical "second round" doesn't match the hashpower (or something related) of those same nodes/pool during the "first round", in order to prevent the situation you're mentioning?

That or programmatically determine if a node is trying to mine in a round they're not allowed to.

Sounds more and more complicated to me as we try to dig deeper Maybe not a good sign.

I'll post the link if I can re find the talk and there are english subtitles. I'm certain "delegated" was a word he mentioned.

I was watching an online talk (in french) by J.-J. Quisquater, a cryptographer mentioned in the bitcoin whitepaper, he was talking about its consensus mechanism.

From what I've understood, he thinks that every miners shouldn't have to mine 24/24, he thinks that was an "overlook".
Instead, it would be more efficient that miners instead reach a consensus in a first round in order to "delegate" in a way to a small set of miners the task/right to actually compete for the block in a second round, the actual competition.
Miners that failed to be part of that second round would "pause" until the next round.

If there is a way to make sure this first round is conducted very fast and securely, then bitcoin could be made more efficient without compromising security.

I was wondering if there's something to dig here? Theoretically is there a way to make bitcoin more energy efficient? Is he right or is there something that Quisquater failed to take into account? If so, what is it?

He's a very serious cryptographer and I don't think claims like this are made lightly. It wasn't a random claim he made, but something he seemed genuinely convinced about. He gave no technical details, it wasn't a technical talk.