Ivrii's conjecture on periodic billiard points

Let $\Omega\subset\mathbb{R}^2$ be a plane region with smooth boundary, and consider the billiard in $\Omega$. The periodic points are the points of the billiard phase space corresponding to periodic billiard trajectories. Ivrii's conjecture. The set of periodic points of the billiard in $\Omega$ has measure $0$.

This conjecture is motivated by Weyl's second-order spectral asymptotics. Ivrii proved that the Weyl conjecture holds when the set of periodic points has measure zero, but the measure-zero assertion itself remains a central open problem in billiards.