A linear bound for real critical points of Fermat hypersurfaces

Let $F_{n,d}$ be the Fermat hypersurface, and let $u\in\mathbb{R}^{n+1}$ be a point for which the squared distance function has finitely many critical points. The real critical points are the real solutions of the critical equation for the squared distance function on $F_{n,d}$.

Linear-bound conjecture. The number of real critical points of the critical equation is at most

$$2n-1.$$

The preceding theorem gives a bound independent of $d$, while the sharp bound is known only in the case $n=1$, where there is one real solution. For $n=2$ the maximum appears to be $3$, but no proof is known, and higher-dimensional cases remain open.