Riley–Wielenberg conjecture on commensurable hyperbolic knots

Let $K$ be a hyperbolic knot, and let $C(K)$ denote the set of knots commensurable with $K$.

Riley–Wielenberg conjecture.

$$|C(K)|\leq 3.$$

This conjecture has been verified in several cases, including all hyperbolic two-bridge knot complements, $(-2,3,n)$-pretzel knot complements, and an infinite family of hyperbolic knot complements. The claim concerns how many knot complements can occur in a commensurability class; its general status is resolved according to the supplied source evidence.