Stabilization conjecture for the expected number of critical tensor approximations

Let $X$ be the tensor rank-one approximation variety, let $d_v$ be the distance function determined by a Gaussian tensor $v$, and let $n_1,\ldots,n_p$ denote the dimensions of the tensor factors. Suppose that

$$n_p-1>\sum_{i=1}^{p-1} n_i-1.$$

Stabilization conjecture. In the Gaussian setting of Theorem 1, the expected number of critical points of $d_v$ on $X$ does not decrease when $n_p$ is replaced by $n_p-1$.

For $p=2$, this follows because a sufficiently general $n_1\times n_2$ matrix with $n_1<n_2$ has $n_1$ singular values, and this number remains unchanged after replacing $n_2$ by $n_2-1$. Over $\mathbb{C}$, the statement is known with equality, but a direct geometric argument explaining the corresponding real statement remains open. The smallest open case is $p=3$ with $(n_1,n_2,n_3)=(2,2,4)$, where the conjecture becomes the explicit integral inequality given in the source statement.