Largest-denominator conjecture for queens

Let $F_q$ be the $q$th Fibonacci number, and consider vertex configurations of $q$ queens in the square-board inside-out polytope. Queen maximal-denominator conjecture. The largest denominator that appears in any vertex configuration for $q$ queens is

$$F_q.$$

The discrete Fibonacci spiral supplies a vertex with denominator $F_q$; the unproved part is that no vertex has a larger denominator.