Isolated critical curvature points for general hypersurfaces

Let $n\geq 4$, and let $X\subset \mathbb{R}^n$ be a general algebraic hypersurface of degree $d$. A critical curvature point is a complex point of the curvature variety at which the curvature polynomial has a critical point.

Isolation conjecture. The hypersurface $X$ has isolated complex critical curvature points.

For surfaces in $\mathbb{R}^3$, the paper proves isolated complex critical curvature points and gives an upper bound for their number. The analogous assertion in dimensions $n\geq 4$ is supported by experiments, but the paper does not provide a proof.