The lattice theta-series root-index conjecture

Let $\Theta_{\Lambda}(x)$ be the theta series of a $d$-dimensional lattice, and let $\mathcal{P}_n$ denote the class of generating functions whose $n$th roots have integral coefficients. The lattice theta-series root-index conjecture. If

$$\Theta_{\Lambda}(x)\in \mathcal{P}_n,$$

then $n\le d$. The authors further conjecture that $\Theta_{\Lambda}(x)\in\mathcal{P}_n$ implies that $n$ divides $d$. The stated bound is motivated by examples suggesting that the values of $n$ in the preceding theorems are best possible for the primes $2$, $3$, $5$, and $7$, but neither this bound nor the stronger divisibility assertion is proved in the source.