The strict descent conjecture for convex polygons

Let $\Omega$ be a convex polygon. Given any two distinct Schwarz reflections $g_i$ and $g_j$, and any $z\in S_{i,j}$ such that $\nabla g_i(z)$ is parallel to $\nabla g_j(z)$, the strict descent condition is the inequality

$$\nabla g_i(z) \cdot \nabla g_j(z) < 0.$$

Strict descent conjecture. Every convex polygon satisfies the strict descent condition. The condition is introduced to construct electrostatic skeletons where different Schwarz reflections coincide. The paper reports that no counterexample is known and proves that polygons satisfying the condition admit electrostatic skeletons with piecewise-analytic support consisting of at most $2n-3$ analytic curves, but does not prove the conjecture for all convex polygons.