Nonpositive-curvature obstruction conjecture for the moduli orbifold

For $g\geq 2$, let $\M_g$ be the moduli orbifold of genus-$g$ Riemann surfaces. Nonpositive-curvature conjecture. The orbifold $\M_g$ admits no complete, finite-volume Riemannian metric with nonpositive sectional curvatures uniformly bounded away from $-\infty$. The source describes this as a basic open problem and discusses possible strengthenings involving finite covers, weaker volume conditions, and ${\rm CAT}(0)$ metrics.