Non-ellipses have caustics with more than four cusps

Let $C$ be an oval, meaning a smooth strictly convex closed curve in the plane, and let an $n$-th caustic by reflection from an interior point $O$ be the envelope of billiard trajectories after $n$ reflections from $C$. Non-ellipse cusp conjecture. If $C$ is not an ellipse, then there exists an integer $n\geq 1$ and an open set $U$ inside $C$ such that, for every $O\in U$, the number of cusps of the $n$-th caustic by reflection from $O$ is greater than four. This is presented as a refined version of an earlier conjecture; the paper establishes the general lower bound of four cusps for generic interior points, but the stated non-ellipse assertion remains open.