Uniform boundedness conjecture for equal-volume Dehn fillings

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold. Uniform boundedness conjecture. There exists a constant $c=c(\mathcal{M})$ such that, for every volume $v$, the number of hyperbolic Dehn fillings of $\mathcal{M}$ having volume $v$ is at most $c$.

This conjecture would answer the paper's questions about the number of hyperbolic Dehn fillings of a fixed volume. The source presents it as a consequence of the preceding multi-cusped conjecture, but supplies no proof of that conjecture or of this consequence.