Containment conjecture for intersections of homoclinic classes

Let $X$ be a Venice mask supported on a compact $3$-manifold $M$, and let $H_1$ and $H_2$ be two different homoclinic classes in $M(X)$. Set $\Lambda=H_1\cap H_2$. Write $\operatorname{Sing}(X)$ for the singularities of the flow and $W^u(\operatorname{Sing}(X))$ for their unstable manifolds; $Cl(\,\cdot\,)$ denotes closure. Containment conjecture.

$$\Lambda\subset Cl(W^u(\operatorname{Sing}(X))).$$

This would follow if the alternative in which a stable-manifold point has alpha-limit contained in the hyperbolic set $H$ could be excluded. The source gives no resolution status.