Ghazouani's Morse–Smale conjecture for dilation surfaces

Let $S$ be a dilation surface that is not a translation surface, and let $\theta\in S^1$ be a direction. A directional foliation is Morse-Smale if it has finitely many closed leaves and the $c9$- and $b1$-limit set of every regular leaf is a closed leaf. Ghazouani's conjecture. For a full-measure set of directions in $S^1$, the directional foliation on $S$ is Morse-Smale.

This conjecture predicts that Morse-Smale behaviour is generic among directional foliations on dilation surfaces that are not translation surfaces. The source notes that it is known for the Disco surface, while no counterexample is known in general.