The generic-path conjecture for traversing vector fields

Let $X$ be a compact smooth manifold with boundary, and let $v_0$ and $v_1$ be traversing vector fields satisfying the properties stated in Corollary 5.2, in particular boundary genericity with $\partial_3X(v_i)=\emptyset$.

Generic-path conjecture. Given $v_0$ and $v_1$, there is a one-parameter family of traversing fields $\{v_t\}_{t\in[0,1]}$ connecting $v_0$ to $v_1$ such that $\partial_3X(v_t)\neq\emptyset$ for only finitely many values of $t$, and for each exceptional value one has $\partial_4X(v_t)=\emptyset$.

This conjecture describes a generic one-parameter interpolation in which codimension-three boundary strata occur only at isolated parameters and higher degeneracies do not occur there. The supplied text gives no proof or resolution.