The maximal-Abelian-subgroup conjecture for SIC Galois groups

Let $\Pi$ be a strongly-centred fiducial, let $S(\Pi)$ be its symmetry group, and let $F\in S(\Pi)$ be such that the unitary $U_F$ is canonical order $3$. Let $\mathbb{E}_1/\mathbb{E}_0$ be the field extension associated with the fiducial, and let $d'$ be the modulus used for the extended Clifford action.

Maximal-Abelian-subgroup conjecture.

$$\operatorname{Gal}(\mathbb{E}_1/\mathbb{E}_0) \cong \mathcal{M}/S(\Pi),$$

where $\mathcal{M}$ is a maximal Abelian subgroup of $\operatorname{GL}(2,\mathbb{Z}/d'\mathbb{Z})$ containing $F$.

The conjecture is stated as a generalization of the type-$z$ relation and is reported to hold for every known exact fiducial, but no general proof is given.