The totally geodesic subvariety conjecture for universal Jacobian loci

Let $J_{g,h}$ be the universal Jacobian locus in $\mathcal A_{g,h}$, equipped with the Riemannian metric $ds^2_{g,h;A,B}$. Universal totally geodesic subvariety conjecture. For a sufficiently large integer $g$, the locus $J_{g,h}$ cannot contain a non-trivial totally geodesic subvariety inside $\mathcal A_{g,h}$ for this metric. The conjecture is posed as the universal analogue of the corresponding statement for $J_g$; its status is not resolved in the supplied text.