The billiard-flow deviation conjecture for rational-angle Euclidean polygons

A rational-angle Euclidean polygon is a Euclidean polygon whose angles are rational multiples of $pi$. The deviation of ergodic averages refers to the asymptotic deviation of time averages for the billiard flow from their ergodic averages. To each such polygon is associated a quadratic differential, with an orbit under $SL(2,\mathbb{R})$.

Billiard-flow deviation conjecture. For all rational-angle Euclidean polygons, the deviation of ergodic averages for the billiard flow is the same for almost all directions, and depends only on the $SL(2,\mathbb{R})$ orbit of the associated quadratic differential.

The conjecture proposes that the fine deviation behavior of billiard trajectories is determined by the $SL(2,\mathbb{R})$ orbit of the associated quadratic differential, extending the connection between Teichmüller-geodesic recurrence and deviations of ergodic averages. The excerpt does not state whether this conjecture has been resolved.