Douglas's asymptotic Kac-polynomial conjecture

Let $\Gamma$ be a quiver with vertex set $\Gamma_0$, let $\R \Gamma_0$ be the associated real vector space, let $\N \Gamma_0$ be the monoid of dimension vectors, and let $a_{\Gamma,n\alpha}(1)$ denote the value at one of the Kac polynomial for the dimension vector $n\alpha$. Douglas's asymptotic Kac-polynomial conjecture. There exists a continuous function $f:\R \Gamma_0\to\R$ such that

$$f(\alpha)=\lim_{n\to\infty}\frac{\ln(a_{\Gamma,n\alpha}(1))}{n}$$

for all dimension vectors $\alpha\in\N \Gamma_0$. This conjecture generalizes an earlier conjecture to arbitrary dimension vectors and concerns the asymptotic exponential growth rate of the Kac polynomial at one. The supplied text attributes it to M. Douglas and gives no resolution.