The slope-set conjecture for taut foliations of knot complements

Let $K$ be a knot in $S^3$ with Seifert genus $g>0$. Let $\mathcal{S}_K\subset\mathbb{Q}$ be the set of rational slopes $s$ for which there exists a taut $\mathcal{C}^0$-foliation on $S^3\setminus K$ meeting the boundary of a tubular neighborhood of $K$ transversally in a nonsingular foliation of slope $s$. The slope-set conjecture. One has

$$\mathcal{S}_K=\mathbb{Q},\qquad \mathcal{S}_K=(-\infty,2g-1)\cap\mathbb{Q},\qquad\text{or}\qquad \mathcal{S}_K=(-2g+1,+\infty)\cap\mathbb{Q}.$$

This is motivated by the $L$-space conjecture and Heegaard Floer calculations for Dehn surgeries on knots in $S^3$. The source notes that it is not known whether $\mathcal{S}_K$ is an interval in general.