Link-volume slope finiteness conjecture for one-cusped hyperbolic manifolds

Let $X$ be a complete finite volume hyperbolic manifold with one cusp. For a slope $\alpha$ on $\partial X$, let $X(\alpha)$ denote the closed manifold obtained by filling $X$ along $\alpha$.

Link-volume slope finiteness conjecture. For any $V>0$, there exists a finite set of slopes $\mathcal{F}$ on $\partial X$, so that if $\lv[X(\alpha)] < V$, then $\alpha$ intersects some slope in $\mathcal{F}$ at most $V/2$ times.

This conjecture asserts that Dehn fillings of a fixed one-cusped hyperbolic manifold with bounded link volume must lie within a controlled intersection distance of a finite collection of slopes. It is presented as a precise formulation of the expectation that link volume grows much faster than hyperbolic volume; no resolution is supplied here.