The second Chern class conjecture for simple vector bundles on abelian surfaces

Let $A$ be an abelian surface, and let $F$ be a simple vector bundle on $A$ with Mukai vector $v:=v(F)$. Suppose that $c_1(F)$ is symmetric. For a positive integer $d$, define

$$S_d(A):=\{z\in \operatorname{CH}_0(A):z+\iota(z)\equiv z'+2m o_A\text{ with }m\in\mathbb Z\text{ and }z'\text{ effective of degree }2d\},$$

where $o_A\in A$ is the origin, $\iota$ is the $(-1)$-involution, and $\equiv$ denotes rational equivalence. Let $d(v)$ denote the integer associated with the Mukai vector $v$ in this filtration. The second Chern class conjecture. Then

$$c_2(F)\in S_{d(v)-1}(A).$$

This conjecture predicts that the second Chern class of a simple vector bundle with symmetric first Chern class lies in the corresponding piece of the filtration on the Chow group of zero-cycles. Its status is not determined by the supplied source context.