Pentagon on an ellipse: maximizing ratio of perimeters

Question

Let $\mathcal{E}$ be an ellipse in the Euclidean plane.

Take 5 points on $\mathcal{E}$, forming a convex pentagon $\mathcal{P}$.

Denote by $\mathsf{Perim}()$ the perimeter.

It seems that $\mathsf{Perim}(\mathcal{P}) \leq \frac{\pi}{5}\cot\left(\frac{\pi}{5}\right) \mathsf{Perim}(\mathcal{E})$ with equality only when $\mathcal{E}$ is a circle and $\mathcal{P}$ is regular.

Any proof or reference of this, if true, or counter-example if false?

Answer 1

This is false, if I am to trust GeoGebra. The ellipse in figure below has semiaxes of lengths $3$ and $\sqrt5$. The pentagon has a side $DF$ of length $3$, perpendicular to the major axis of the ellipse, sides $DE$, $FC$ parallel to the major axis, and the fifth vertex $C$ at one of its endpoints.