Can the hash of UTXO set be part of the bitcoin block header and consensus rules and in which cases?

This question doesn’t have much logic from the “reality” perspective, it’s more theoretical about whether something can or cannot be part of consensus rules and in which cases.

I know that the consensus rule can only be something that all nodes see the same way. In other words, at all points in time all nodes must come to the same conclusion about some chain (whether its valid or invalid).

I wonder if the hash of some set of UTXOs can be part of the header and consensus rule? I have two questions related to it:

1. Can we set as a consensus rule that some hash from the header block must be equal to the hash of strictly node’s current UTXO set?
2. Can we set as a consensus rule that some hash from the header block must be equal to the hash of UTXO set that exists in current point in chain (the set of unspent outputs that currently exists in the chain from the previous block to genesis)?
3. Can we set as a consensus rule that some hash from the header block must be equal to the hash of some UTXO set (does not need to be nodes current UTXO set (case 1) nor UTXO set that exist in current point of chain (case 2))? I literally mean some “random” set of UTXOs that we would get, for example, within a block (like we get transactions) or not even within a block but some that we must find then by ourselves.