Zilber–Pink density conjecture for real multiplication in low genus

Let $K$ be a totally real number field of degree $g$, and define

$$E_{g,K}:=\{x\in\mathcal{M}_g:\operatorname{End}(J_x)^0\cong K\}.$$

Low-genus density conjecture. If $g\leq 6$, then for each $K$, $E_{g,K}$ is dense in $\mathcal{M}_g(\mathbb{C})$. This is the complementary low-genus prediction to the high-genus finiteness statement; the supplied text gives no resolution status.