Minimal-perimeter conjecture for ordinary reduced hyperbolic polygons

Let $P\subset H^2$ be an ordinary reduced $n$-gon of minimal width $w$. Let $\widetilde{P}$ denote the regular ordinary reduced $n$-gon of width $w$, and write $\operatorname{perim}(P)$ for the perimeter of $P$. Minimal-perimeter conjecture.

$$\operatorname{perim}(P)\leq\operatorname{perim}(\widetilde{P})$$

with equality if and only if $P$ is regular. The conjecture proposes that, among ordinary reduced hyperbolic $n$-gons with a fixed minimal width, the regular polygon has maximal perimeter; the surrounding discussion explains that the usual Jensen-inequality argument for the Euclidean and spherical cases does not apply in the hyperbolic plane.