The billiard decomposition conjecture for planar domains

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with smooth boundary, and let $p=1,2,\dots$. A billiard trajectory is a trajectory in $\Omega$ satisfying the billiard reflection law at boundary points. The billiard decomposition conjecture. There is a finite set of billiard trajectories $\gamma_{p,1},\dots,\gamma_{p,N(p)}\subset\Omega$, possibly with repetition, such that

$$\omega_p(\Omega)=\sum_{j=1}^{N(p)}\operatorname{length}(\gamma_{p,N(p)}).$$

This is the natural planar-domain analogue of the result that widths of closed surfaces are sums of lengths of closed geodesics. The source describes the claim as well known to experts, but the supplied text gives no resolution status.