Avoidance conjecture for weakly strict saddle points

Let $(M,g)$ be a real-analytic Riemannian manifold, and let $f:M\to\mathbb{R}$ be real-analytic and non-constant. For a subset $C\subset\operatorname{Crit}(f)$, write $W^s_-(C)$ for the set of points whose negative-time trajectories approach $C$. A point is a weakly strict saddle point when it satisfies the paper's weak strict-saddle condition.

Avoidance conjecture. If a compact connected component $C$ of $\operatorname{Crit}(f)$ consists of weakly strict saddle points, then $W^s_-(C)$ has measure zero. Moreover, if $M$ is compact and $C\subset\operatorname{Crit}(f)$ is any compact subset, then $W^s_-(C)$ has measure zero.

The conjecture would extend the paper's avoidance results, including the weak center-stable theorem and its consequences, beyond the tameness assumption. Its status is not determined by the supplied text.