The general-type conjecture for high modular-tower levels

Let $\mathbb H$ be a modular tower, with levels parametrized by $k$, and let $\mathbb Q$ denote the rational field. A variety is of general type when some multiple of its canonical bundle gives a projective embedding. General-type conjecture for high modular-tower levels. High modular-tower levels have general type and no $\mathbb Q$-points. The claim is part of the proposed obstruction to regular inverse Galois realizations at high levels; the source gives no resolution.