The type-a Galois-group conjecture for SIC fiducials

Let $\Pi$ be a strongly-centred type-$a$ fiducial, let $S(\Pi)$ be its symmetry group, and let $F\in S(\Pi)$ be such that $U_F$ is a canonical order $3$ unitary. Let $d'$ be the relevant modulus, and choose a matrix $G$ satisfying

$$F=I+3G.$$

Define

$$\mathcal{A}=\{rI+sG:r,s\in\mathbb{Z}/d'\mathbb{Z}\text{ and }rI+sG\in\operatorname{GL}(2,\mathbb{Z}/d'\mathbb{Z})\}.$$

Type-a Galois-group conjecture.

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

This is identified in the source as a weaker conjecture proposed previously for strongly-centred type-$a$ fiducials; the stronger maximal-Abelian-subgroup formulation is reported to hold in all known exact cases.