The planar robustness conjecture for regular polygons

For $n>2$, let $\rho_n$ denote the maximal robustness among planar convex solids in the equilibrium class with $n$ points, and let $\rho($regular n-gon$)$ denote the robustness of the regular $n$-gon. Planar robustness conjecture.

$$\rho_n=\rho(\text{regular n-gon})$$

for $n>2$. The conjecture is motivated by the external and internal robustness results for regular polygons; its resolution is not supplied here.