The index-vanishing conjecture for commuting vector fields on 3-manifolds

Let $M$ be a $3$-manifold, and let $X$ and $Y$ be $C^1$ commuting vector fields on $M$. Let $U$ be a compact subset of $M$ such that

$$\operatorname{Zero}(Y)\cap U=\operatorname{Zero}(X)\cap\partial U=\emptyset.$$

Index-vanishing conjecture. Then

$$\operatorname{Ind}(X,U)=0.$$

This extends the stated analytic result in dimension three and the corresponding lower-dimensional regularity result; the supplied context presents it as the motivation for the conjecture, with no resolution given here.