Multiplet count conjecture for Weyl–Heisenberg SICs

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, set $\Delta=\Delta_d=(d+1)(d-3)$ and $K=\Q(\sqrt{\Delta})$. Let $\mathcal O_\Delta$ and $\mathcal O_K$ denote the quadratic order of discriminant $\Delta$ and the maximal order of $K$, respectively. Multiplet Count Conjecture. The number of Galois multiplets of Weyl–Heisenberg SICs in dimension $d$ equals the number of quadratic orders $\mathcal O'$ satisfying $\mathcal O_\Delta\subseteq\mathcal O'\subseteq\mathcal O_K$, equivalently $\sigma_0(f)$ when $\Delta=f^2\Delta_0$ with $\Delta_0$ fundamental. This predicts the number of Galois multiplets but not their internal geometric class counts; it remains open.