Douglas's asymptotic Euler-characteristic conjecture for Kronecker moduli spaces

Let $m\geq 3$ and let $M^m_{d,e}$ denote the Kronecker moduli space for the $m$-arrow Kronecker quiver and dimension vector $(d,e)$. Set

$$m_1=\frac{m-\sqrt{m^2-4}}{2},\qquad m_2=\frac{m+\sqrt{m^2-4}}{2}.$$

Douglas's conjecture. There exists a continuous function $f:[m_1,m_2]\subset\mathbb{R}\to\mathbb{R}$ such that, for every $r\in[m_1,m_2]$ and every $\varepsilon>0$, there are $\delta>0$ and $n\in\mathbb{N}$ for which every coprime $(d,e)\in\mathbb{N}^2$ satisfying $|r-e/d|<\delta$ and $|d+e|>n$ also satisfies

$$\left|f(r)-\frac{\ln(\chi(M^m_{d,e}))}{d}\right|<\varepsilon.$$

The conjecture asserts that the normalized logarithm of the Euler characteristic is asymptotically determined by the slope $e/d$ and varies continuously with that slope. The paper presents a precise formulation based on Douglas's conjecture; the supplied text gives no evidence that this formulation has been resolved.