The Neuwirth conjecture for knots

Let $K$ be a non-trivial knot in $S^3$.

Neuwirth conjecture. There exists a closed surface $F$ such that $F$ contains $K$ non-separatingly and is essential in the exterior of $K$.

The conjecture is known for alternating, generalized alternating, Montesinos, and several other classes of knots, but remains open in general.