The intersection-map conjecture for surfaces with $p_g=1$ and $q=2$

Let $S$ be a surface with geometric genus $p_g(S)=1$ and irregularity $q(S)=2$. Let $A^1_{hom}(S)_{\mathbb{Q}}$ be the group of homologically trivial divisor classes with rational coefficients, and let $A^2_{AJ}(S)_{\mathbb{Q}}$ be the Albanese kernel with rational coefficients. The intersection map is

$$A^1_{hom}(S)_{\mathbb{Q}}\otimes A^1_{hom}(S)_{\mathbb{Q}}\longrightarrow A^2_{AJ}(S)_{\mathbb{Q}}.$$

Assume that the image of the Albanese map has dimension less than $2$. The intersection-map conjecture. The intersection map is zero.

The preceding argument shows that surjectivity would contradict the induced map on $H^1(S,\mathcal O_S)$ and $H^2(S,\mathcal O_S)$; the stronger assertion that the map is zero is stated conjecturally in this setting.