The boundary-generic realization conjecture for traversing fields

Let $X$ be a compact connected smooth manifold of dimension $n+1\geq 3$ with a traversing vector field $v$. Let $Z^+$ and $Z^-$ be disjoint, nonempty, closed $(n-1)$-submanifolds of $\partial X$ whose union separates $\partial X$ into two $n$-manifolds $Y^+$ and $Y^-$. Let $h:X\to\operatorname{int}(X)$ be an orientation-preserving diffeomorphism, and let boundary-genericity and the strata $\partial_i^\pm$ have the meanings specified in the paper.

Boundary-generic realization conjecture. The constraints

$$\chi(Y^+)=\chi(X),\qquad n\equiv0\pmod 2,$$

and

$$\chi(Z^+)-\chi(Z^-)=2\chi(X),\qquad n\equiv1\pmod 2,$$

are necessary and sufficient for the existence of such an $h$ for which the restriction of $v$ to $h(X)$ is boundary generic, $\partial_1^\pm(h(X))(v)=h(Y^\pm)$, $\partial_2^\pm(h(X))(v)=h(Z^\pm)$, and $\partial_3(h(X))(v)=\emptyset$. Moreover, in any given collar $U$ of $\partial X$ in $X$, such an $h$ can be chosen $U$-supported and arbitrarily close to the identity in the $C^0$ topology.

The Euler-characteristic conditions are shown in the supplied text to be necessary. Sufficiency is described as potentially requiring an h-principle argument, so the conjecture remains open on the supplied evidence.