The rationality conjecture for brane-tiling Donaldson–Thomas generating functions

Let $G$ be a consistent brane tiling, let $(Q,W)$ be the corresponding quiver with potential, and let $A=\mathbb{C}Q/(\partial W)$ be its quiver-potential algebra. Fix a vertex $i\in Q_0$, and write $r=\# Q_0$. Let $Z^i(A)$ denote the associated generating function, and let $\Log$ be the plethystic logarithm. Rationality conjecture. The specialization

$$\left.\Log\bigl(Z^i(A)\bigr)\right|_{x_1=\dots=x_r=x}\in\mathbb{Q}[[x]]$$

is a rational function. This conjecture predicts rationality after identifying all vertex variables in the plethystic logarithm of the noncommutative Donaldson–Thomas generating function. It is motivated by product formulas established for orbifolds of $\mathbb{C}^3$ and for the conifold, but the supplied source does not state a general resolution.