Gordon's purely cosmetic surgery conjecture in the 3-sphere

Let $K$ be a nontrivial knot in $S^3$. For surgery slopes $r$ and $s$, the manifolds $S^3_r(K)$ and $S^3_s(K)$ are purely cosmetic when they are homeomorphic as oriented manifolds. Gordon's purely cosmetic surgery conjecture. If $r eq s$, then

$$S^3_r(K) \ncong S^3_s(K).$$

No examples of purely cosmetic surgeries are known, and the conjecture asserts that distinct surgery slopes on a nontrivial knot never yield homeomorphic oriented manifolds.