Bloch-type conjecture on intersections of homologically trivial divisors

Let $S$ be a smooth projective surface over $\mathbb{C}$. Write $A^j(S)$ for the Chow group of codimension-$j$ cycles modulo rational equivalence, let $A^1_{\mathrm{hom}}(S)$ denote homologically trivial divisors, and let $A^2_{\mathrm{AJ}}(S)$ denote the subgroup of Abel–Jacobi trivial zero-cycles. Consider the cup product map

$$H^1(S,\mathcal O_S)\otimes H^1(S,\mathcal O_S)\to H^2(S,\mathcal O_S).$$

Bloch-type conjecture. If this cup product map is zero, then the intersection product map

$$j_S\colon A^1_{\mathrm{hom}}(S)\otimes A^1_{\mathrm{hom}}(S)\to A^2_{\mathrm{AJ}}(S)$$

is also zero. This conjecture predicts that vanishing of the cup product in coherent cohomology forces all intersections of homologically trivial divisors to vanish in the Chow group of zero-cycles. The paper proves the conjecture for Sicilian surfaces.