Very ampleness bound conjecture for generalized theta linear series

Let $X$ be a smooth projective curve, and let $SU_X(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$, with generalized theta line bundle $\mathcal{L}$. Very ampleness bound conjecture. The linear series $|\mathcal{L}^m|$ is very ample for every $m\geq r^2+r$, for arbitrary rank and degree. The bound is known to give very ampleness when $r$ and $d$ are coprime, while the source notes that the general case is obstructed by the unresolved problem of separating tangent vectors at singular points of the moduli space.