Persistent foliar branched surface conjecture for non-L-space knots

Let $K$ be a knot in $S^3$ that is not an L-space knot, meaning that no nontrivial Dehn surgery on $K$ yields an L-space. A persistently foliar branched surface in the exterior of $K$ is a branched surface whose induced structures give taut foliations on all nontrivial Dehn fillings of the exterior. Persistent foliar branched surface conjecture. The exterior of every non-L-space knot in $S^3$ has a persistently foliar branched surface.

This is motivated by announced constructions for alternating and Montesinos knots without L-space surgeries and by the paper’s results for nonalternating knots. Its general status is not resolved in the supplied text.