The monotonicity and divisibility conjecture for coefficient periods

Let $u_{\mathbb P}(q;n)$ be the quasipolynomial for a piece $\mathbb P$, and write $p_i$ for the period of its coefficient $\gamma_i$. Coefficient-period monotonicity conjecture. The periods satisfy

$$1=p_0=p_1\leq\cdots\leq p_{2q},$$

and, more precisely,

$$p_i\mid p_{i+1}.$$

The conjecture is suggested by the nested systems of subspaces used to define the denominators $D_i^q$, since $D_i^q$ divides $D_{i+1}^q$ and Ehrhart theory gives $p_i\mid D_i^q$. The paper does not establish the claimed monotonicity in general.