Minimal-period conjecture for coefficients of restricted-partition quasi-polynomials

Let $a_1,a_2,\ldots,a_n$ be positive integers, and let

$$r(x)=\sum_{k\geq 0}q(k)x^k=\frac{1}{(1-x^{a_1})(1-x^{a_2})\cdots(1-x^{a_n})},$$

where $q(k)=c_d(k)k^d+\dots+c_0(k)$ is a quasi-polynomial. For a positive integer $m$, define $b_m=\#\{i:m\mid a_i\}$. For $0\leq j\leq d$, define

$$p_j=\operatorname*{\lcm}\{m:b_m>j\}.$$

Minimal-period conjecture. The minimum period of $c_j(k)$ is $p_j$ for every $0\leq j\leq d$. This conjecture predicts that the divisibility data of the $a_i$ determine the exact periods of all coefficient functions in this class of generating functions. The preceding discussion presents it as a more general statement that would imply the conjectured extension of the quasi-polynomial convolution theorem; its resolution is not given in the source.