The regular polygon convergence conjecture for Möbius and hyperbolic metrics

Let $G_k$ be the regular $k$-gon described in the paper, and let $x,y\in\bigcap_k G_k$ be distinct points. Let $\rho_{G_k}$ and $\delta_{G_k}$ denote the Möbius and hyperbolic metrics in $G_k$. The regular polygon convergence conjecture.

$$\lim_{k\to\infty}\frac{\delta_{G_k}(x,y)}{\rho_{G_k}(x,y)}\to1.$$

The conjecture expresses convergence of the metric quotient toward the equality known in the unit disk as regular polygons increasingly resemble the disk; no proof or resolution is stated.