Four-cusp conjecture for caustics by reflection in ellipses

Let $C$ be an ellipse and let $O$ be an interior point that is not a focus of $C$. For each $n\geq 1$, the $n$-th caustic by reflection from $O$ is the envelope of the family of billiard trajectories after $n$ reflections from $C$. Four-cusp conjecture. For all $n\geq 1$, this caustic has exactly four cusps, and all four are ordinary cusps. This refines the known lower bound of four cusps for the caustics by reflection from a generic interior point of an oval; the $n=1$ case without the ordinary-cusp assertion is related to Jacobi's Last Geometric Statement, while the general claim remains open.