Conjecture on rigid classes and cyclic commensurability of hyperbolic knots

A commensurability class of hyperbolic $3$-manifolds is called rigid when its unique minimal element has a single cusp whose horospherical cross-section is a Euclidean turnover. Two knot complements are cyclically commensurable if they have a common finite cyclic cover. Cyclic commensurability conjecture. A rigid commensurability class does not contain cyclically commensurable hyperbolic knot complements.

The preceding argument constructs distinct cyclically commensurable knots in a commensurability class arising from an orbi-lens space surgery. The conjecture asserts that this phenomenon cannot occur in a rigid class; the source does not provide a resolution.