Ellipses characterized by four-cusp caustics

Let $\gamma$ be a planar billiard table that is not an ellipse. For a light source chosen inside $\gamma$, let $\Gamma_n$ be the caustic by reflection after $n$ reflections, with $n\geq 1$. Four-cusp characterization conjecture. For some choice of the light source inside $\gamma$ and some $n\geq 1$, the caustic $\Gamma_n$ has more than four cusps. This is the second conjecture presented in the paper and is described there as potentially over-optimistic; no resolution is supplied.