Mukai's conjecture on semi-homogeneous presentations of stable sheaves

Let $X$ be an abelian surface with $\operatorname{NS}(X)=\mathbb{Z} H$ and let $v$ be a Mukai vector with

$$\ell:=\langle v^2\rangle/2>0.$$

Suppose that $v$ has at least one solution of the numerical equation. Among the numerical solutions for $v$, choose $(v_1,v_2,\ell_1,\ell_2)$ such that $\operatorname{rk}(v_i)$ is minimum, where $i\in\{1,2\}$ is determined by $\ell_i=\ell$. Let $\mathfrak{M}^{+}(v_1,v_2,\ell_1,\ell_2)$ be the moduli space of the corresponding simple two-term complexes, and let $M_X^H(v)$ be the moduli space of $H$-stable sheaves with Mukai vector $v$. Mukai's conjecture. For a general member $[E^{-1}\xrightarrow{f}E^0]$ of $\mathfrak{M}^{+}(v_1,v_2,\ell_1,\ell_2)$, the morphism $f$ is either surjective or injective; in that situation, its kernel or cokernel is stable; and a general member of $M_X^H(v)$ has a semi-homogeneous presentation corresponding to this numerical solution. This conjecture predicts that stable sheaves on an abelian surface satisfying the stated numerical condition arise from presentations by semi-homogeneous sheaves. The source attributes it to Mukai; the supplied text gives no resolution status.