Persistence conjecture for critical sets with one fast and two slow variables

Let $g$ be a fast vector field with one fast variable and two slow variables, and let

$$\mathcal{C}[g]=\{p=(x,y)\in\mathbb{R}\times\mathbb{R}^2:g(p)=0\}$$

be its critical set. Let

$$\mathcal{D}[g]=\mathcal{D}_1[g]\cup\mathcal{D}_2[g]\cup\mathcal{D}_3[g]\cup\mathcal{D}_4[g]\cup\mathcal{D}_5[g]\cup\mathcal{D}_6[g]$$

be the union of the listed degenerate subsets of the fold set, and let $V_f$ denote the relevant space of fast vector fields.

Persistence conjecture. For one fast and two slow variables, the critical set $\mathcal{C}[g]$ is persistent under perturbations for $g\in V_f$ if all folds are non-degenerate, that is, if $\mathcal{D}[g]=\emptyset$.

This extends the analogous persistence criterion from the one-fast, one-slow case. The source states it as a conjecture and gives no resolution.