Order-to-multiplet conjecture for Weyl–Heisenberg SICs

Fix a positive integer $d\neq 3$, let $\Delta_d=(d+1)(d-3)$, and write $\Delta_d=f^2\Delta_0$, where $\Delta_0=\operatorname{disc}(\mathcal O_{K_d})$ and $K_d=\Q(\sqrt{\Delta_d})$. Let $\mathcal O'$ range over the intermediate quadratic orders, indexed by positive divisors $f'$ of $f$, and let $\mathcal M(f')$ denote a Galois multiplet of Weyl–Heisenberg line-SICs. Order-to-Multiplet Conjecture. There is a bijection between the positive divisors $f'$ of $f$ and the Galois multiplets, such that the multiplet indexed by $f'$ contains $|\operatorname{Cl}(\mathcal O_{(f')^2\Delta_0})|$ geometric equivalence classes, and $f_1\mid f_2$ implies $\fieldvec{S_1}\subseteq\fieldvec{S_2}$ for $[S_i]\in\mathcal M(f_i)$. This predicts a precise organization of SIC multiplets by quadratic orders; the stronger version additionally imposes compatibility for the associated triple-product and projection fields.