Arithmetic conjecture for dual expected internal-angle sums

Let $n\in\mathbb{N}$ and $k\in\{1,\ldots,n\}$. Let $\beta>(n-1)/2$ be an integer or half-integer, and let $\tilde{\mathbb{J}}_{n,k}(\beta)$ denote the corresponding dual expected internal-angle sum. Arithmetic conjecture. If both $2\beta-n$ and $k$ are even, then

$$\tilde{\mathbb{J}}_{n,k}(\beta)= \begin{cases} q\pi^{-(n-k)}, & n-k\text{ is even},\\ q\pi^{-(n-k-1)}, & n-k\text{ is odd}, \end{cases}$$

for some rational number $q$. This is a symbolic-computation-based strengthening of the preceding arithmetic theorem and is not proved in the source.