Cyclotomic Hilbert-series conjecture for finite-dimensional graph algebras

Let $G$ be a graph for which the associated algebra $\mathcal E_G$ is finite-dimensional, and let $\mathcal H_G(t)$ denote its Hilbert series.

Cyclotomic Hilbert-series conjecture. The Hilbert series $\mathcal H_G(t)$ is a product of cyclotomic polynomials.

This conjecture proposes a uniform form for the Hilbert series whenever $\mathcal E_G$ is finite-dimensional. It is presented as an observed pattern, and the supplied text gives no proof or resolution.