Contractibility conjecture for weighted barycentric spaces

Let $\Sigma$ be a compact surface with singular points $p_1,\ldots,p_m$ and weights $-1<\alpha_1\leq\cdots\leq\alpha_m<0$. For $\rho>0$, let $\Sigma_{\rho,\underline{\alpha}}$ denote the space of formal barycenters associated with the singular Liouville problem, and call it $p_1$-stable when adjoining $p_1$ to every admissible support still gives total weight below $\rho/(4\pi)$. Topological version. The space of formal barycenters $\Sigma_{\rho,\underline{\alpha}}$ is contractible if and only if it is $p_1$-stable. The conjecture would classify when the formal-barycenter space is contractible, complementing the existence theorem based on its non-contractibility; the paper presents examples and motivates an equivalent algebraic criterion, but does not establish the converse in general.