Kac's constant-term conjecture for loopless quivers

Let $\Gamma=(\Gamma_0,\Gamma_1)$ be a finite quiver with no loops, let $\alpha\in\mathbb{N}^n$ be a dimension vector, and let $A_\Gamma(\alpha,q)$ be its Kac polynomial. Let the Kac-Moody algebra associated to $\Gamma$ have root vector $\alpha$ with a specified multiplicity. Kac's conjecture. The constant term of $A_\Gamma(\alpha,q)$ is equal to the multiplicity of the root vector $\alpha$ in the associated Kac-Moody algebra. This conjecture was proved by Crawley-Boevey and Van den Bergh for indivisible dimension vectors and by Hausel for arbitrary dimension vectors.