Injectivity conjecture for the second Albanese map

Let $X$ be a projective nonsingular surface over ${\bold C}$, and let $T(X)$ be the kernel of the Albanese map on degree-zero zero-cycles. Let $\underline{{\mathrm{M}}}({\bold C})$ be the category of arithmetic Hodge structures used in the paper, and let $\rho_X^2$ be the second Albanese map. Second Albanese injectivity conjecture. The map

$$\rho_X^2:T(X)\longrightarrow {\mathrm{Ext}}^2_{\underline{{\mathrm{M}}}({\bold C})}({\bold Q}(0),H^2(X)(2))$$

is injective for every projective nonsingular surface $X$, without assuming $p_g=0$. This strengthens the relationship between algebraic zero-cycles and higher extensions in arithmetic Hodge structures. The source presents the assertion as a conjecture and gives no resolution.