The convex polygonal-domain Möbius–hyperbolic inequality conjecture

Let $G\subsetneq\mathbb{R}^2$ be a bounded convex polygonal domain, and let $\rho_G$ and $\delta_G$ denote the Möbius and hyperbolic metrics in $G$. The convex polygonal-domain inequality conjecture. There exist points $x,y,u,v\in G$ such that

$$\rho_G(x,y)<\delta_G(x,y)\qquad\text{and}\qquad \rho_G(u,v)>\delta_G(u,v).$$

The claim is motivated by computations for an equilateral triangle and asserts that neither metric inequality holds uniformly even in bounded convex polygonal domains; no resolution is given.