The real quadratic ray class field conjecture for SICs in every dimension

Let $d\ge 4$ be a dimension, and let $\mathfrak{R}$ denote the real quadratic ray class field associated with the dimension. For a SIC orbit $E$ in $\mathbb{C}^d$, let $\mathbb{Q}(E)$ be the smallest subfield of $\mathbb{C}$ containing the projective fields of definition of all its lines.

Real quadratic ray class field conjecture. The statement that, for the dimensions listed in Proposition 1, there is a SIC orbit $E$ with $\mathbb{Q}(E)=\mathfrak{R}$, while every other known orbit has $\mathbb{Q}(E)/\mathbb{Q}$ a finite Galois extension containing $\mathfrak{R}$, holds for every dimension $d\ge 4$.

This conjecture extends the observed ray-class-field pattern from the explicitly known dimensions to all dimensions at least four. Its status is unresolved in the supplied source.