Conjectured bijection between SIC-POVMs and ring class monoid elements

Prove the conjectured bijection between SIC-POVMs in dimension d and the elements of the ideal class monoid \(\Clmpa(\mathcal{O})\) for the quadratic order \(\mathcal{O}=\mathbb{Z}[\frac{d-1+\sqrt{(d+1)(d-3)}}{2}]\).

Background

The paper introduces several ray class monoids for orders in number fields, including $\Clmpa(\mathcal{O})$, which denotes the ordinary ideal class monoid obtained by allowing non-invertible ideal classes. In the concluding discussion, the author connects these monoids to SIC-POVMs, configurations of $d^2$ equiangular complex lines in $\mathbb{C}^d$.

The cited claim is presented as a conjectural correspondence: for the specified quadratic order depending on $d$, SIC-POVMs are expected to be parametrized by elements of $\Clmpa(\mathcal{O})$. The paper does not establish this bijection, so determining or proving it remains unresolved within the paper.