Concavity conjecture for maximal-area disk-polygons

Let $C$ be a circle of radius $r<1$, and let $P_n$ be an $n$-sided disk-polygon of largest area inscribed in $C$.

Concavity conjecture for maximal-area disk-polygons. For all $n\geq 4$,

$$\operatorname{Area}(P_{n-1})+\operatorname{Area}(P_{n+1})<2\operatorname{Area}(P_n).$$

The paper proves this inequality for odd $n$ with $n\geq 5$, but states that its method does not extend straightforwardly to all $n$ and that no general area formula is known. The conjecture extends the established odd-index case to every $n\geq4$.