The conjectured sharp perimeter bound for convex decagons

Let $P$ be a convex $10$-gon in the Euclidean plane, let $\Gamma$ be its boundary, let $L(\Gamma)$ denote the length of $\Gamma$, and let $\delta(\Gamma)$ denote its self Chebyshev radius.

Decagon perimeter conjecture. One has

$$L(\Gamma)\geq 20\left(1+\sqrt{5}-\sqrt{5+2\sqrt{5}}\right)\cdot\delta(\Gamma).$$

This is a sharp-looking extremal inequality suggested by calculations and computer experiments on convex polygons. The supplied text gives no resolution status.