There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not Zariski dense.
Notably, for $g>>0$ the moduli space $M_g$ of genus $g$ curves is general type (Harris-Mumford proved this for $g\geq 24$). For some past discussion of Lang's conjecture on $M_g$ see What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves? and links within.
Now, one can construct rational curves in $\overline{M_g}$ by taking a base-point-free pencil in a linear system on a smooth surface. Conversely, every rational curve in $\overline{M_g}$ whose generic point has trivial stabilizer arises this way, by resolving the total space of the associated curve fibration.
The case I am interested is when this surface is rational: For every choice of a smooth surface $S$ and a base-point-free pencil on $S$ whose generic element has genus $g$, consider the associated rational curve in $\overline{M_g}$. Can one prove unconditionally that for $g >> 0$ there is some nontrivial Zariski-closed subset of $\overline{M_g}$ containing all these rational curves?
