Characterizing slopes conjecture for knots

Let $K$ be a knot in $S^3$. A rational number $p/q$ is a characterizing slope for $K$ if an orientation-preserving homeomorphism between $S^3_K(p/q)$ and $S^3_{K'}(p/q)$ implies that $K$ and $K'$ are isotopic. Here $S^3_K(p/q)$ denotes the $p/q$-surgery on $K$.

Characterizing slopes conjecture. There exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 2$ and $|p|+|q|\geq C$ is characterizing for $K$.

This gives a precise formulation of the expectation that, for every knot, almost all non-integer slopes are characterizing. The general statement remains open, although it is known for several individual knots and infinite classes of knots.