The strengthened billiard-trajectory conjecture for rounded disk-polygons

Let $D$ be a fat disk-polygon in the Euclidean plane, and let $D(\varepsilon)$ be its $\varepsilon$-rounded disk-polygon, obtained as the union of all circular disks of radius $\varepsilon$ contained in $D$. A generalized billiard trajectory is a billiard path in this rounded disk-polygon, and a $2$-periodic trajectory has period two.

Rounded disk-polygon billiard conjecture. Any of the shortest generalized billiard trajectories in $D(\varepsilon)$ is a $2$-periodic one for all $\varepsilon$ at most as large as the inradius of $D$.

The paper states this as a stronger version of a theorem valid for all sufficiently small positive $\varepsilon$. The stronger assertion is presented as a belief rather than a proved result, so its resolution is not supplied here.