The Cabling Conjecture for reducible surgeries on knots in $S^3$

Let $K$ be a knot in $S^3$. A surgery on $K$ is reducible if the resulting $3$-manifold contains an essential $2$-sphere, meaning a $2$-sphere that does not bound an embedded $3$-ball. The cabling annulus of a cabled knot is the annulus associated with its cable structure and determines a surgery slope.

Cabling Conjecture. If $K$ is a knot in $S^3$ which has a reducible surgery, then $K$ is a cabled knot and the reducing slope is given by the cabling annulus.

All known examples arise from $pq$-surgery on the $(p,q)$-cable $C_{p,q}(K)$, for which $S^3_{pq}(C_{p,q}(K))\cong L(p,q)\#S^3_{q/p}(K)$. The conjecture asserts that these are the only reducible surgeries on knots in $S^3$ and remains open in general.