The Galois-width conjecture for Gaussian-mixture moment maps

Let $\pi_{e,k}$ be the moment map for a $k$-component Gaussian mixture model using the first $e$ moments, and let $\operatorname{gw}(\pi)$ denote the Galois width of the corresponding branched cover. For every integer $k\ge 1$,

Galois-width conjecture.

$$\operatorname{gw}\left(\pi_{3k-1,k}\right)=\frac{\deg(\pi_{3k-1,k})}{k!}.$$

The statement refines algebraic identifiability by predicting that the Galois width of the $(3k-1)$-moment map is its degree divided by the label-swapping factor $k!$. The surrounding discussion says that algebraic identifiability is known, while the related rational-identifiability statement for $3k$ moments remains open; it does not provide a resolution of this Galois-width equality.