Conway-sphere conjecture for L-space knots

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. An essential Conway sphere is an essential $4$-punctured sphere in the knot complement. Conway-sphere conjecture. An L-space knot contains no essential Conway spheres in its complement. This is proposed as a structural restriction on L-space knots and remains open in the source.