Kac–Moody $L^2$ Euler characteristic conjecture for parabolic character varieties

Let $\bar{\M}^\mu_\B$ be the character variety quotient associated with a genus-$g$ Riemann surface and parabolic type $\mu$, and let $\chi_{L^2}(\bar{\M}^\mu_\B)$ denote its topological $L^2$ cohomology Euler characteristic. Let $(\Gamma,\vv)$ be the associated star-shaped quiver and dimension vector, and let $m_\vv$ be the multiplicity of the weight $\vv$ in the Kac--Moody algebra $\mathfrak{g}(\Gamma)$. The $L^2$ Euler characteristic conjecture.

$$\chi_{L^2}(\bar{\M}^\mu_\B)=\begin{cases}0,&g>1,\\1,&g=1,\m_\vv,&g=0.\end{cases}$$

This is presented as a consequence of the purity conjecture and connects the topology of character varieties with the Kac denominator formula. Since the supplied text gives no independent resolution of this consequence, it is recorded as open.