Can a hash of a UTXO set be part of the Bitcoin block header and consensus rules? And in what cases?

This question doesn’t have much logic from a “reality” perspective. It is more theoretical about whether and under what circumstances some sets of UTXOs can be part of the consensus rule.

The consensus rule knows that only what all nodes see equally is possible. That is, at every point in time, every node must reach the same conclusion about some chain (valid or not).

I was wondering if the hash of the UTXO set could be part of the header and consensus rules. I have two questions:

1. Could there be a consensus rule that the new hash of the header block must be the same as the hash of the current UTXO set?
2. Could there be a consensus rule that the new hash of the header block must be equal to the hash of some set of UTXOs (it doesn’t have to be strictly a set of UTXOs)? SOME literally means the set of UTXOs that we either get within a block (like getting a transaction) or don’t get within a block. Then we’ll have to find it ourselves, but again the rules are: It doesn’t have to be that way. Same as the current UTXO set.