Bor–Tabachnikov conjecture on cusps of elliptical billiard caustics

Let $\gamma\subset\mathbb{R}^2$ be an ellipse, let $O$ be a light source inside $\gamma$ and different from a focus, and for each $n\geq 1$ let $\Gamma_n$ be the envelope of rays from $O$ that have undergone $n$ reflections in $\gamma$. Bor–Tabachnikov conjecture. For every $n\geq 1$, the caustic by reflection $\Gamma_n$ has exactly four ordinary cusps. The conjecture is the billiard analogue of Jacobi's conjecture on cusps of caustics from points on ellipsoids. It is known that the caustics have at least four ordinary cusps for general ovals and that $\Gamma_1$ has exactly four ordinary cusps, while the assertion for all $n\geq 1$ remains open.