The Cabling Conjecture for hyperbolic knots in lens spaces

Let $K$ be a knot in a lens space, and let surgery on $K$ produce a non-prime $3$-manifold $Y$. A knot is hyperbolic if its exterior admits a complete finite-volume hyperbolic structure.

Baker's conjecture. If $K$ is hyperbolic, then

$$Y=L(r,1)\mathbin{\#}L(s,1).$$

Otherwise, either $K$ is a torus knot, a Klein bottle knot, or a cabled knot and the surgery is along the boundary slope of an essential annulus in the exterior of $K$, or $K$ is contained in a ball.

This conjecture generalizes the Cabling Conjecture from $S^3$ to lens spaces by proposing that Baker's family accounts for all hyperbolic knots with non-prime surgeries. The statement was disproved by a counterexample constructed in the paper.