Critical-point count conjecture for heterogeneous quadratics with two vectors

Consider the heterogeneous quadratics minimization problem

$$\min_{Z^TZ = {\rm Id}_k} \sum_{i=1}^k Z_i^T A_i Z_i,$$

where $A_i \in {\rm Sym}(\mathbb{R}^n)$ and $Z_i$ is the $i$th column of $Z$. Critical-point count conjecture. For $k=2$, the number of critical points is

$$8 \sum_{j=1}^{n-1} j^2.$$

Counting the complex critical points of this optimization problem is described as a challenging open problem; the displayed formula is supported by numerical computations for small values of $n$ and $k$.