Geometric count conjecture for Weyl–Heisenberg SICs

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, let $\Delta=(d+1)(d-3)$, and let $\mathcal O_\Delta$ be the quadratic order of discriminant $\Delta$. Geometric Count Conjecture. The number of geometric equivalence classes of Weyl–Heisenberg covariant SICs is $|\operatorname{WHSIC}(d)/\mathcal{PEC}(d)|=|\operatorname{ClM}(\mathcal O_\Delta)|$. This connects the enumeration of SICs up to geometric equivalence with the class monoid of a quadratic order; it remains open in general.