C+: If one believes that one has conclusive reason to have A, then one is rationally required to have A; and
C-: If one believes that one lacks sufficient reason to have A, then one is rationally required not to have A.
Now consider someone who believes that they have conclusive reason to do what they believe they lack sufficient reason to do. (Granted, this is a very odd belief to have. But I think it is possible. Perhaps they've been told that their beliefs have been manipulated by an evil demon into being unreliable [update: this is explained further in my comments below]. Or perhaps they're just incredibly irrational. Whatever.)
It would then follow from C+ that they are rationally required to do what they believe they lack sufficient reason to do. Call this action 'X'. That is, we have so far established that they are rationally required to X. But recall that our agent believes that they lack sufficient reason to X. It thus follows from C- that they are rationally required not to X. Putting these two results together, we find that our poor confused agent is both rationally required to X and rationally required not to X!
This violates what I will call the "consistency of rational requirements" principle:
(CRR) It is not possible for one to be both rationally required to A, and rationally required not to A.
In other words, rationality cannot make contradictory demands of us. It cannot demand both that we do something, and that we don't do it. That's just not a fair ask.
If (CRR) is true, as I think it is, then the case I provide above shows that Kolodny's "core requirements" cannot be true.