Conjecture on distinct critical points for two-dimensional linear Gaussian covariance models

Let $A$ and $B$ be generic matrices defining the two-dimensional linear Gaussian covariance model $\mathcal{M}_{A,B}$, and let $S$ be a generic sample covariance matrix. The score equations are the likelihood critical-point equations for this model.

Distinct-solutions conjecture. For generic values of $A$, $B$, and $S$, the score equations of $\mathcal{M}_{A,B}$ with sample covariance matrix $S$ have $2n-3$ distinct solutions.

The proven ML-degree is $2n-3$, so the conjecture asserts that all solutions are distinct rather than occurring with higher multiplicity. The source also raises the separate open question of how many of these solutions can be real; that question is not included as a conjecture row here.