Appleby–Flammia–McConnell–Yard ray class field conjecture for SICs

Let $d\geq 4$, set $\Delta_d=(d+1)(d-3)$, and let $K_d=\Q(\sqrt{\Delta_d})$. For a Weyl–Heisenberg line-SIC $S$ in dimension $d$, let $\fieldvec{S}$ denote its SIC field, and let $H_{d'\infty_1\infty_2}$ be the ray class field of $K_d$ with modulus $d'\infty_1\infty_2$, where $d'=d$ for odd $d$ and $d'=2d$ for even $d$. Appleby, Flammia, McConnell, and Yard's conjecture. At least one Weyl–Heisenberg line-SIC has $\fieldvec{S}=H_{d'\infty_1\infty_2}$, and every such line-SIC has $\fieldvec{S}$ a finite Galois extension of $\Q$ containing $H_{d'\infty_1\infty_2}$. The two assertions were verified in the cited dimensions and for all currently known SICs there, but remain conjectural in general.