Monotonicity conjecture for the function $F_q$

Let $n\in\mathbf N$, let $q\geq 1$, and let $F_q$ be the function defined in the preceding analysis. Denote by $[a]$ the integer part of $a$.

Monotonicity conjecture. $F_q$ is decreasing on $[0,\frac{\pi}{2}]$ for $q>2$. For $q\leq 2$, $F_q$ is nondecreasing on $[0,\frac{\pi}{2}]$ when $\left[\frac{(n+1)q}{2}\right]$ is even, and nonincreasing on $[0,\frac{\pi}{2}]$ when $\left[\frac{(n+1)q}{2}\right]$ is odd.

The conjecture is motivated by the fact that $\beta=0$ and $\beta=\frac{\pi}{2}$ are stationary points of $F$ for $q\geq 1$ and $n\in\mathbf N$, together with numerical experiments. The supplied text gives no proof or resolution of the general monotonicity claims.