Factorization conjecture for nonstandard slow-fast vector fields

Let $H(z,\varepsilon)$ be a vector field satisfying Assumptions --, with slow manifold $S_{\varepsilon}$ and linear fast fibers $\mathcal{N}_{\varepsilon}(p)$ at basepoints $p\in S_{\varepsilon}$. The tangent space to the slow manifold at $p$ is denoted by $T_pS_{\varepsilon}$. Factorization conjecture. Such vector fields may be re-factorized as

The new objects satisfy $S_{\varepsilon}=\{z\in\mathbb{R}^n:f_{\varepsilon}(z,\varepsilon)=0\}$; the columns of $N_{\varepsilon}(p,\varepsilon)$ span $\mathcal{N}_{\varepsilon}(p)$; and $\widetilde{G}(p,\varepsilon)\in T_pS_{\varepsilon}$ for every $p\in S_{\varepsilon}$. The conjecture would provide a geometric factorization in which the slow manifold is a level set, the columns of $N_{\varepsilon}$ encode the fast fibers, and the remainder is tangent to the slow manifold. The source presents this as a conjecture and gives no resolution, so its status remains open.