Generic Morse–Smale conjecture for directional foliations on dilation surfaces

Let $X$ be a dilation surface that is not a translation surface, and let $m\in\mathbb{RP}^1$ be a direction. The directional foliation in direction $m$ is the foliation by geodesic-flow leaves of slope $m$; a foliation is Morse–Smale if it has finitely many closed leaves such that the $\alpha$- and $\omega$-limit sets of every leaf are among them.

Generic Morse–Smale conjecture. The collection of directions $m\in\mathbb{RP}^1$ for which the directional foliation of $X$ is Morse–Smale has full measure.

This conjecture predicts that, for a generic direction on a non-translation dilation surface, the geodesic flow is attracted to finitely many closed leaves. The supplied source does not state whether the conjecture has been resolved.