Gordon's conjecture on Seifert fibered surgeries on hyperbolic knots

Let $K\subseteq S^3$ be a hyperbolic knot, and let $S^3_{p/q}(K)$ denote the result of $p/q$-Dehn surgery on $K$. A surgery is integer when $q=1$.

Gordon's conjecture. If $S^3_{p/q}(K)$ is a Seifert fibered space, then

$$q=1.$$

This conjecture asserts that Seifert fibered manifolds cannot arise from non-integer surgeries on hyperbolic knots in $S^3$. It would replace the exceptional-surgery results used in the paper and would imply the subsequent characterization of Montesinos knots with proper rational unknotting number one.