The Geometric Conjecture on general type of Modular Tower components

Let a Modular Tower be associated to a $p$-perfect centerless group $G_0$ and the relevant collection of conjugacy classes, and let its level-$k$ spaces have nonsingular projective completions of their components. Geometric Conjecture. For $k$ sufficiently large, the nonsingular projective completions of all components of a Modular Tower at level $k$ have general type. This geometric formulation is presented as a version of the Main Conjecture and concerns the eventual birational complexity of every component; the source does not state that it has been proved or disproved.