Kotěšovec's Fibonacci-period conjecture for queens

Let $F_q$ be the $q$th Fibonacci number, let $[F_q]=\{1,2,\ldots,F_q\}$, and let $u_{\mathbb Q}(q;n)$ be the counting quasipolynomial for $q$ queens. Kotěšovec's conjecture. The period of $u_{\mathbb Q}(q;n)$ is

$$\operatorname{lcm}[F_q],$$

the least common multiple of all positive integers up through $F_q$. Kotěšovec's conjectured periods through $q=7$ agree with this formula, but the general assertion is not proved in the supplied text.