Convex polygon billiard wave-front density conjecture

Let a convex polygon billiard be a billiard in a convex polygonal table, and let $W_t(P)$ denote the billiard wave front at time $t$ from a point $P$ in the table. The wave front becomes dense if it eventually intersects every open subset of the billiard table. Convex polygon-billiard density conjecture. For all convex polygon billiards, all the wave fronts $W_t(P)$ become dense. The source describes this as a problem that may admit a simple argument but has not yet been settled, so the conjecture remains open.