Zilber–Pink conjecture for real multiplication loci in the Torelli locus

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

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

Real multiplication finiteness conjecture. Fix $g\geq 7$. Each $E_{g,K}$ is contained in a finite union of Hilbert modular varieties. Moreover, $E_{g,K}$ is empty for all but finitely many $K$. This predicts that, in high genus, real multiplication occurs in only finitely many relevant Hilbert-modular families and for only finitely many totally real fields; the supplied text gives no resolution status.