Galois-field conjecture for Weyl–Heisenberg SICs

Fix $d\geq1$, $d\neq3$, and let $S$ be a $d$-dimensional Weyl–Heisenberg line-SIC with associated fields $\fieldtrip{S}$, $\fieldproj{S}$, and $\fieldeproj{S}$. Galois-field conjecture. The vectors of $S$ are projective-algebraic, and $\fieldtrip{S}$, $\fieldproj{S}$, and $\fieldeproj{S}$ are Galois extensions of $\Q$. The conjecture extends the algebraicity prediction by asserting normality of the associated fields; it remains open, although part of it follows conditionally from the earlier ray class field conjecture.