Vertex attainment conjecture for the distance-to-conformal-radius ratio

Let $P$ be a bounded convex $n$-gon, decomposed into regions $P_j$ consisting of points closer to the $j$-th side than to the other sides. Let $Gr(P)$ be the resulting graph after removing the boundary points of $P$ and the open edges having nonempty intersection with $\partial P$. For $u\in P$, write $d_P(u)$ for the distance to the boundary and $r_P(u)$ for the conformal radius.

Vertex attainment conjecture. The maximum

$$\max_{u\in P}\frac{2d_P(u)}{r_P(u)}$$

is attained at a vertex of the graph $Gr(P)$. This is stronger than the preceding theorem, which only guarantees attainment at some point of $Gr(P)$. The conjecture concerns sharp localization of the extremum for convex polygonal domains and remains unresolved in the supplied source.