Kac's conjecture for coherent sheaves on weighted projective lines

Let $\mathbb{X}_{\mathbf{p},\underline{\lambda}}$ be a weighted projective line, let $\mathfrak{Lg}$ be its corresponding loop Kac–Moody algebra, and use the identification $K_0(\operatorname{Coh}(\mathbb{X}_{\mathbf{p},\underline{\lambda}}))\simeq\mathfrak{h}^*_{\mathfrak{Lg}}$. For a coherent sheaf $\mathcal{F}$, write $[\mathcal{F}]$ for its class. Kac's conjecture for weighted projective lines. The following hold: (i) there exists an indecomposable coherent sheaf $\mathcal{F}\in\operatorname{Coh}(\mathbb{X}_{\mathbf{p},\underline{\lambda}})$ if and only if $[\mathcal{F}]\in\mathfrak{h}^*_{\mathfrak{Lg}}$ is a positive root; moreover, the set of such indecomposables forms a $1-\frac{(\alpha,\alpha)}{2}$-parameter family with a unique component of maximal dimension; (ii) for every root $\alpha$ there exists a polynomial $P_\alpha(v)\in\mathbb{N}[v]$ with leading term $v^{1-\frac{(\alpha,\alpha)}{2}}$ such that the number of absolutely indecomposable coherent sheaves of dimension $\alpha$ over $\mathbb{F}_q$ is $P_\alpha(q)$, and $P_\alpha(0)=\operatorname{dim}\mathfrak{Lg}_\alpha$. The supplied text presents this as a variant of Kac's quiver conjectures, but gives no resolution status.