McMullen's Arithmetic Chaos conjecture

Let $\mathcal{X}=T^1(\Gamma\backslash\mathbb{H})$ be the unit tangent bundle of the modular surface. A closed geodesic is defined over a real quadratic field $K$ when its associated arithmetic data are defined over $K$.

McMullen's Arithmetic Chaos conjecture. There is a compact subset $\mathcal{Y}\subset\mathcal{X}$ such that, for every real quadratic field $K$, the set of closed geodesics defined over $K$ and lying in $\mathcal{Y}$ has positive entropy.

McMullen posed this as an arithmetic analogue of a dynamical-chaos problem concerning closed geodesics that remain in a compact part of the modular surface. The supplied text gives no resolution.