Positive topological entropy conjecture for lemon billiards

Let $\mathcal{Q}(b)$ be the intersection of two unit disks with center distance $b\in(0,2)$, and let

$$F_b:M_b\to M_b$$

be the billiard map on the lemon table $\mathcal{Q}(b)$. Lemon billiard entropy conjecture. For every $0<b<2$,

$$h_{top}(F_b)>0.$$

The conjecture asks whether every lemon billiard has positive topological entropy; the paper proves this for $b$ in a one-sided neighborhood of $1.5$, apart from possibly a discrete subset, by establishing crossing homoclinic and heteroclinic intersections.