The vanishing conjecture for reduced regulators on self-products of very general surfaces

Let $X=X/{\mathbb C}\subset {\mathbb P}^3$ be a very general surface of degree $d\geq 5$. The reduced regulator map is

$$\underline{r}_{3,1}:\operatorname{CH}^3(X\times X,1)\to H_{\operatorname{tr}}^4(X\times X,{\mathbb R})\bigcap H^{2,2}(X\times X),$$

where $H_{\operatorname{tr}}^4(X\times X,{\mathbb R})$ is the space of transcendental cocycles.

Reduced-regulator vanishing conjecture. The map $\underline{r}_{3,1}$ is zero.

This predicts that the reduced real regulator detects no nontrivial transcendental $(2,2)$-classes on the self-product of a very general surface of degree at least five. The source presents this as an expectation, and no resolution is supplied here.