I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized version of the same statement):
Theorem: Let $f : M \rightarrow Q$ be a fibered over $B$ map covered by a fibered epimorphism $F : T_BM \rightarrow T_BQ$, where $T_BX$ denotes the vertical tangent space of $X$ over $B$. Suppose that $F$ coincides with $df$ near a closed subset $K \subset M$ (so that $f$ is a fibered submersion near $K$), then there exists a fibered wrinkled map $g : M \rightarrow Q$ which extends $f$ from a neighborhood of $K$, and such that the fibered epimorphisms $\mathcal{R}(dg)$ and $F$ are homotopic rel. $T_BM|_K$.
Here, $\mathcal{R}(dg)$ is a canonical regularization of the differential $dg$ which doesn't vanish near the critical points of $g$, so that $\mathcal{R}(dg)$ is a fiberwise epimorphism covering $g$. Using this result, they go on and prove a generalization of Igusa's theorem in their second paper.
My question is the following: how technical are these assumptions to verify for families of Morse functions? Namely, for a $k$-parameter family of Morse functions $f:M \times D^k \rightarrow \mathbb{R} \times D^k$ fibered over $D^k$, can one always find a covering epimorphism, perhaps by somehow regularizing the differential as one does for wrinkles? Since the output wrinkle $g$ of the above theorem is $C^0$-close to $f$, this seems equivalent to asking for conditions on $f$ which guarantee a $C^0$-approximation by wrinkled functions. I have looked for applications of this theorem to particular maps, but I haven't found quite what I'm looking for. In addition, it seems like Theorems 2.3, 2.4, and 2.5 of their second paper are relevant to this question.
Any references or thoughts are greatly appreciated.
