Effectivity of the jumping divisor for curves in the moduli space

Let $f:T\to\overline{\mathcal M}_g$ be a morphism from a projective curve whose image is not contained in $\Delta_0$, and let $J_T$ denote the associated jumping divisor. Weak jumping-divisor conjecture. The divisor $J_T$ is effective. This would strengthen Moriwaki's inequalities for curves meeting the singular part of $\Delta_0$ by giving a non-negative contribution from the jumping divisor. The source does not state a resolution.