The general-type and rational-point conjecture for high modular-tower levels

Let $K$ be a number field and let $\{\mathcal H_k\}_{k=0}^{\infty}$ be a modular tower, meaning a sequence of Hurwitz-space levels associated with a fixed Nielsen class and its successive group levels. A level has a $K$-point if it has a point rational over $K$.

High-level modular-tower conjecture. At sufficiently high levels, there are no $K$-points on a modular tower. Moreover, sufficiently high levels are algebraic varieties of general type, so a sufficiently high power of their canonical bundle embeds each such level in projective space.

This is identified in the source as the Main modular-tower conjecture and is attributed to Fried (1995).