The hyperbolic closed geodesic conjecture for strict dilation surfaces

Let $X$ be a dilation surface, meaning a surface equipped with an atlas whose transition maps have the form $z\mapsto az+b$ with $a\in\mathbb{R}_{+}^{\ast}$ and $b\in\mathbb{C}$. A dilation surface is strict if it is not a translation surface, and a closed geodesic is hyperbolic if its monodromy representation has a nontrivial dilation ratio.

The hyperbolic closed geodesic conjecture. Every strict dilation surface contains a hyperbolic closed geodesic.

This conjecture was proposed by DFG. The paper presents it as the first of three conjectures of increasing strength; the supplied text gives no resolution.