The classification conjecture for hyperbolic doubly primitive knots in non-prime manifolds

Let $K$ be a doubly primitive knot, meaning a knot of the type specified by the paper's doubly primitive construction, in a non-prime $3$-manifold. Let $(p,q)$ be a surgery slope, and let $K_{r,s}^{p,q}$ denote the paper's knot associated to integers $r,s$.

Doubly primitive knot classification conjecture. If $K$ is hyperbolic, then

$$K=K_{r,s}^{p,q}$$

for some slope $(p,q)$ and integers $r,s\neq\pm1$.

The conjecture records the authors' failure to find other hyperbolic doubly primitive examples: attempts to generalize known sporadic constructions generally produced surgery duals of cabled knots or knots with Seifert-fibered exteriors. The claim is left as a direction for further investigation.