Tautologicality conjecture for curves on Enriques surfaces

Let $E$ be an Enriques surface, and let $C\subset E$ be an irreducible nonsingular curve of genus $g\geq 2$. Its moduli point in $\overline{\mathcal{M}}_g$ determines a $0$-cycle. Tautologicality conjecture. This $0$-cycle is tautological. The conjecture concerns the image of curves on Enriques surfaces in the moduli space of curves and predicts that their associated zero-cycles lie in the tautological ring.