Direction-independent stabilization conjecture for normalized Kac polynomials

Let $Q=(I,\Omega)$ be a quiver, let $\operatorname{\mathbf{d}}\in\operatorname{\mathbf{N}}^I$, and let $v_{\underline{n}}$ denote the valuation of $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)$. The normalized Kac polynomials are viewed as elements of the formal power-series ring $\operatorname{\mathbf{N}}[[q]]$.

Stabilization conjecture. The sequence

$$\frac{A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)}{q^{v_{\underline{n}}}}\in\operatorname{\mathbf{N}}[q]$$

converges in $\operatorname{\mathbf{N}}[[q]]$ as $\underline{n}\to(+\infty,\ldots,+\infty)$, and its limit is the power-series expansion at $0$ of a rational fraction.

The conjecture is equivalent to independence of the limiting normalized polynomial from the direction along which the arrow multiplicities tend to infinity. The paper presents it as an expected strengthening of the directional convergence result.