Polynomiality conjecture for the ML degree of linear covariance models

Let $m$ be the dimension of a linear covariance model and let $n$ be its ambient dimension. The ML degree is the number of complex critical points of the likelihood equations for a generic model, counted with multiplicity.

ML-degree polynomiality conjecture. For $m\geq 3$, the ML degree is a polynomial of degree $m-1$ in $n$. In the cases stated in the source, for $m=3$ it equals

$$3n^2-9n+7,$$

and for $m=4$ it equals

$$\frac{11}{3}n^3-18n^2+\frac{85}{3}n-15.$$

The conjecture extends the observed polynomiality along the rows of the table of generic models. The case $m=2$ is known separately, where the ML degree equals $2n-3$; the supplied text does not state a resolution for the conjecture for general $m\geq 3$.