The Seifert surgery conjecture for knots in the three-sphere

Let $K$ be a knot in $S^3$, let $p/q\in\mathbb{Q}$ be a surgery slope with $q\geq 2$, and write $S_{p/q}^3(K)$ for the manifold obtained by $p/q$-surgery on $K$. A Seifert fibred surgery is a surgery yielding a Seifert fibred space. Seifert surgery conjecture. If $S_{p/q}^3(K)$ is a Seifert fibred space and $q\geq 2$, then $K$ is either a torus knot or a cable of a torus knot. This conjecture would identify the knots admitting non-integer Seifert fibred surgeries and underlies the proposed classification of non-integer characterizing slopes for torus knots; its resolution is not supplied here.