The characterization of star-shaped trajectories with rotation number two-fifths

Let $D$ be the domain under consideration, and let $\triangle P_1P_2P_3$ be a triangle in $D$. Write $\psi_{\triangle P_1P_2P_3}$ for its associated billiard map, $\rho(\psi_{\triangle P_1P_2P_3})$ for the rotation number, $\delta(P_i,P_j,P_k)$ for the corresponding geometric quantity, and $\Delta_1'(P_i,P_j)$ and $\Delta_2'(P_i,P_j)$ for the two quantities associated with the pair $(P_i,P_j)$. The conjecture. One has $\rho(\psi_{\triangle P_1P_2P_3})=\frac{2}{5}$ if and only if there exist $i,j,k$ with $\{i,j,k\}=\{1,2,3\}$ such that

$$\Delta_2'(P_i,P_j)\leq \delta(P_i,P_j,P_k)\leq \frac{1}{2}\Delta_1'(P_i,P_j)$$

or

$$\frac{1}{2}\Delta_1'(P_i,P_j) \leq \delta(P_i,P_j,P_k)\leq \Delta_2'(P_i,P_j).$$

The theorem cited in the source proves the sufficient condition, and the paper establishes the conjecture for a certain class of isosceles triangles; the necessity in general remains open.