The period-denominator equality conjecture for inside-out polytopes

Let $\mathscr{A}_{\mathbb P}$ be the inside-out arrangement associated with the piece $\mathbb P$, and let $D([0,1]^{2q},\mathscr{A}_{\mathbb P})$ denote the denominator of the corresponding inside-out polytope. Let $D_i^q$ be the least common denominator of the coordinates of all points obtained by intersecting subspaces of codimension at most $i$ in the arrangement with the boundary of $[0,1]^{2q}$. If $p$ is the period of $u_{\mathbb P}(q;n)$ and $p_i$ is the period of its coefficient $\gamma_i$, period-denominator equality conjecture.

$$p=D([0,1]^{2q},\mathscr{A}_{\mathbb P}),\qquad p_i=D_i^q.$$

In general Ehrhart theory a period need not equal the denominator, but the examples discussed in the paper exhibit equality. The asserted equality is presented as a stronger conjecture and remains unproved in general.