Is the union of the complete curves on $\mathcal M\_g$ Zariski dense? ($g \gg 0$)


I know it is hard to find higher-dimensional complete subvarieties of $\mathcal M\_g$, but a quasiprojective variety can have lots of complete curves but nothing higher dimensional, e.g. the complement of a codimension 2 linear subspace of $\mathbb P^n$. I am curious whether the known complete curves can deform in $\mathcal M\_g$.



Locus of complete curves on $\mathcal M_g$
Let $M\_g^S$ be the Satake compactification of $M\_g$. It is a singular projective variety. For $g \geq 3$, the codimension of $M\_g^S \setminus M\_g$ is $\geq 2$. If $p \in M\_g$ is a point, then a sufficiently general linear subspace through $p$ cuts out a complete curve not meeting the Satake boundary. Conclusion: through any point of $M\_g$, $g \geq 3$, passes a complete curve.


