The totally geodesic subvariety conjecture for Jacobian loci

Let $J_g$ be the Jacobian locus in $\mathcal A_g$, and let a totally geodesic subvariety mean an irreducible subvariety arising as the image of a totally geodesic subvariety of the relevant symmetric domain. Totally geodesic subvariety conjecture. For a sufficiently large integer $g$, the Jacobian locus $J_g$ cannot contain a non-trivial totally geodesic subvariety. The conjecture is false for an integer $g\leq 6$ and is open in the sufficiently large-genus range stated.