The Morse-Smale directional-flow conjecture for strict dilation surfaces

Let $X$ be a strict dilation surface, meaning a dilation surface that is not a translation surface. For each direction in $\mathbb{S}^{1}$, consider the corresponding directional flow of $X$.

The Morse-Smale directional-flow conjecture. There is an open dense subset $U$ of $\mathbb{S}^{1}$ such that the directional flow of $X$ along any direction of $U$ is Morse-Smale.

The paper states this as the third of three conjectures of increasing strength and proves, in the abstract, an equivalence between this property and density of the directions of hyperbolic closed geodesics. The supplied text gives no resolution of the conjecture itself.