Lidman–Moore conjecture on tangle primeness of L-space knots

Let $K$ be a knot in a $3$--manifold $Y$. An essential $n$--string tangle decomposition is an embedded sphere $Q$ intersecting $K$ transversally in $2n$ points such that $Q-\partial\mathcal{N}(K)$ is incompressible and boundary-incompressible in the knot exterior $Y-\mathcal{N}(K)$. The knot is $n$--string prime if it has no essential $n$--string tangle decomposition. Lidman–Moore conjecture. L-space knots are $n$--string prime for all integers $n>0$; equivalently, L-space knots have no essential tangle decomposition. This strengthens the conjecture that L-space knots have no essential Conway spheres. The paper presents it as a stronger expectation, and it remains open.