The geodesic noncontainment conjecture for the compactified Jacobian locus

Let $J_g$ be the Jacobian locus in $\mathcal A_g$ and let $\overline J_g$ be its closure. Geodesic noncontainment conjecture. For a sufficiently large integer $g$, there does not exist a geodesic in $\mathcal A_g$ contained in $\overline J_g$ and intersecting $J_g$. This is stated as a stronger version of the totally geodesic subvariety conjecture; no resolution is given in the supplied text.