Log Bloch conjecture for smooth algebraic surfaces

Let $U$ be a smooth algebraic surface, let $p_g(U)$ denote its geometric genus, and let $T(U)$ be the Albanese kernel, namely the kernel of the Albanese morphism $a:h_0(U)^0\to \operatorname{Alb}(U)$. Log Bloch conjecture. If

$$p_g(U)=0,$$

then

$$T(U)=0.$$

When $U$ is projective, this is the classical Bloch conjecture; the paper proves the assertion for open smooth complex surfaces when the Bloch conjecture holds for a compactification, including all $\mathbb{Q}$-homology planes and open smooth surfaces that are not of log general type.