The density conjecture for hyperbolic closed geodesic directions on dilation surfaces

Let $X$ be a strict dilation surface, where a strict dilation surface is a dilation surface that is not a translation surface. A closed geodesic is hyperbolic when its monodromy representation has a nontrivial dilation ratio, and its direction is its slope in the dilation-surface charts.

The density conjecture for hyperbolic closed geodesic directions. The set of directions of hyperbolic closed geodesics of $X$ forms a dense subset of $\mathbb{S}^{1}$.

A weak version, asserting only that directions of closed geodesics are dense in $\mathbb{S}^{1}$, has been proved by BGT; the supplied text says that this does not determine whether a significant part of these geodesics are hyperbolic. The stated conjecture remains unresolved in the supplied text.