The almost-everywhere ergodicity and weak-mixing conjecture for polygonal billiards

Let $P$ be a simply connected polygonal table, and let the billiard flow on its $3$-dimensional phase space be the Hamiltonian motion of a point moving freely in $P$ and reflecting specularly at the boundary.

Polygonal-billiard conjecture. For almost all simply connected polygonal tables, the billiard flow is ergodic and weakly mixing.

The ergodic theory of general polygonal billiards is poorly understood. Although a $G_\delta$-dense set of polygons with ergodic billiard flow is known, the asserted almost-everywhere combination of ergodicity and weak mixing is not established in the source.