On a conjecture of Zhao related to standard relations among cyclotomic multiple zeta values

We provide a proof of a conjecture by Zhao concerning the structure of certain relations among cyclotomic multiple zeta values in weight two. We formulate this conjecture in a broader algebraic setting in which we give a natural equivalence between two schemes attached to a finite abelian group $G$. In particular, when $G$ is the group of roots of unity, these schemes describe the standard relations among cyclotomic multiple zeta values.