Generic stratification conjecture for Pareto critical sets

Let $W$ be a compact manifold of dimension $n\geq m$, let $u\in C^\infty(W,\mathbb{R}^m)$, and let $\theta$ be its Pareto critical set, with $\theta_S$ denoting the singular part of $\theta$. A stratification of a subset of $W$ is a finite collection of connected submanifolds satisfying the frontier and transversality conditions. Generic stratification conjecture. There is an open and dense set $\mathcal{G}\subset C^\infty(W,\mathbb{R}^m)$ such that if $u\in\mathcal{G}$, then $\theta$ is a stratified set and $\theta_S$ is a union of strata. The conjecture concerns the generic geometric structure of Pareto critical sets and is stated as proved only for $m=2,3$; its validity in general remains open.