Delman–Roberts' persistent foliation conjecture for knots in L-spaces

Let $K$ be a knot in an L-space. The knot $K$ is persistently foliar if, except for one meridional slope, every boundary slope of its complement is strongly realized by a co-oriented taut foliation. A surgery is non-trivial if its slope is not the meridional slope. Delman–Roberts' persistent foliation conjecture. The knot $K$ is persistently foliar if and only if it has no non-trivial L-space or reducible surgeries. This is a knot-theoretic version of the L-space conjecture and remains open in the stated generality.