Universal interval conjecture for taut foliations in knot complements

Let $k$ be a nontrivial knot in $S^3$, and let $M=S^3\setminus\operatorname{int}(N(k))$ be its exterior. For slopes $s\in(-a,b)$, require that $M$ have a taut foliation whose restriction to $\partial M$ is a collection of circles of slope $s$, and that attaching disks along those boundary circles extends the foliation to a taut foliation in the Dehn filling $M(s)$. Universal interval conjecture. The maximal interval $(-a,b)$ with this property always contains $(-1,1)$.

The question asks for the maximal interval associated to each nontrivial knot. The source states the containment as a conjectural claim and gives no resolution.