Strong Cabling Conjecture

Let $g\ge 0$, $b\ge 1$, and $q\ge 1$ be integers. A triple $(g,b,q)$ is called realizable if there exist a knot $K\subset S^3$ and a compact orientable essential surface $F\subset E(K)$ of genus $g$, with $b$ boundary components and boundary slope $p/q$ for some integer $p$. For an even integer $b\ge 6$, the triple $(0,b,1)$ is never realized.

Strong Cabling Conjecture. For any even integer $b\ge 6$, the triple $(0,b,1)$ is not realizable.

This conjecture is motivated by known restrictions on realizable triples: triples $(g,1,q)$ with $q\ge 2$ are not realizable, planar surfaces require $q=1$, and the triple $(0,4,1)$ is not realizable. It implies the famous Cabling Conjecture, which asserts that a Dehn surgery on a knot in $S^3$ can yield a reducible manifold only when the knot is a cable knot and the surgery slope is that of the cabling annulus.