The geodesic noncontainment 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 geodesic noncontainment conjecture. For a sufficiently large integer $g$, there does not exist a geodesic contained in $J_{g,h}$ for this metric. The statement is presented as a further conjecture for universal Jacobian loci; no resolution is given in the supplied text.