Arithmetic conjecture for expected internal-angle sums

Let $\beta\geq -1$ be an integer or half-integer, and let $n\in\mathbb{N}$ and $k\in\{1,\ldots,n\}$. The quantity $\mathbb{J}_{n,k}(\beta)$ denotes the expected internal-angle sum associated with these parameters. Arithmetic conjecture. If both $2\beta+n$ and $n-k$ are odd, then

$$\mathbb{J}_{n,k}(\beta)=q\pi^{-(n-k-1)}$$

for some rational number $q$. This strengthens the known arithmetic description in the corresponding odd-parity case; it is based on symbolic computations and is not proved in the source.