Lidman–Moore conjecture on essential Conway spheres in L-space knots

Let $K$ be a knot in $S^3$. An essential Conway sphere is an embedded sphere intersecting $K$ transversally in four points such that the resulting four-punctured sphere is incompressible and boundary-incompressible in the knot exterior. Lidman–Moore conjecture. An L-space knot has no essential Conway sphere. This is a geometric decomposition conjecture for L-space knots; the paper classifies L-space Montesinos knots and finds no counterexamples, but the general claim remains open.