The exact fields present in a JWK is a frequent cause of interoperability problems.
We don't currently appear to have an language (in VCI or HAIP) that says whether kid, use, alg are required to be sent, so issuers & wallets are likely to make different assumption.
I'm not fully thought it through but in the general case people seem to like it when kid is present so we might want to at least mandate that in HAIP.
If we're not going to mandate it then it would probably be good to be clear that implementations need to tolerate kid being absent.
Whatever we do decide to say should probably apply to the jwk in the openid4vci-proof+jwt header.