Commensurability conjecture for hyperbolic knot complements

Let $K$ be a hyperbolic knot, and consider the commensurability class of its complement $S^3\setminus K$. A knot complement is a complement of a knot in $S^3$ that lies in this class. Commensurability conjecture. (i) There are at most three knot complements in the commensurability class of $S^3\setminus K$. (ii) If $K$ does not admit symmetries or hidden symmetries, then there is only one knot complement in the commensurability class of $S^3\setminus K$. The conjecture is motivated by the preceding examples and by the observation that a hyperbolic knot without symmetries or hidden symmetries is expected to be the unique knot complement in its commensurability class. The paper does not establish the asserted universal bound or the uniqueness statement in full generality.