Global generation conjecture for theta line bundles on moduli spaces of degree-zero bundles

Let $X$ be a smooth projective curve of genus $g$, let $U_X(r,0)$ denote the moduli space of semistable vector bundles of rank $r$ and degree zero on $X$, and let $N\in\operatorname{Pic}^{g-1}(X)$. Write $\Theta_N$ for the corresponding theta divisor and $\mathcal{O}_{U}(k\Theta_N)$ for its $k$th tensor power.

Global generation conjecture. For any $r\geq 1$, $\mathcal{O}_{U}(k\Theta_N)$ is globally generated on $U_X(r,0)$ for $k\geq r+1$.

This conjecture proposes a uniform effective base point freeness bound extending the known cases $r=1$, $r=2$, and $r=3$ discussed immediately before the statement. The optimality of the bound for general rank is not established in the supplied text.