Queen inside-out-polytope denominator conjecture

Let $F_q$ be the $q$th Fibonacci number and let $[F_q]=\{1,2,\ldots,F_q\}$. For the inside-out polytope associated with $q$ queens, let $D$ denote its denominator. Queen denominator conjecture. The denominator of the inside-out polytope for $q$ queens is exactly

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

where $[F_q]=\{1,2,\ldots,F_q\}$. This is presented as a weaker denominator analogue of Kotěšovec's period conjecture and is left as an aim for the paper's theory.