The high-level genus conjecture for modular-tower components

Let $\overline{\mathcal H}_k$ be the nonsingular projective closure of a component of the level-$k$ inner reduced Hurwitz space in a modular tower.

High-level genus conjecture. For large $k$, all components of $\overline{\mathcal H}_k$ have genus exceeding $1$.

This assertion is used to imply the weak Main Conjecture in the cited special case, since curves of genus greater than $1$ have strongly restricted rational points. The source gives no resolution of the assertion itself.