The bounded height conjecture in geometric topology

Let $M$ be a $k$-cusped hyperbolic $3$-manifold. A Dehn filling point is a point corresponding to a Dehn filling of $M$, and its height is the height used in the associated Dehn filling parameterization. Bounded height conjecture in geometric topology. The height of any Dehn filling point of $M$ is uniformly bounded. This conjecture was formulated to imply finiteness of hyperbolic $3$-manifolds with bounded volume and trace field degree. The paper's abstract states that the author proves it, so its status is solved.