CTF detection conjecture for non-simple knots in lens spaces

Let $L$ be a lens space and let $K\subset L$ be a non-simple knot. The knot meridian determines a boundary slope of the knot complement, and a slope is CTF-detected if it is realized by the boundary foliation of a co-oriented taut foliation. Non-simple knot CTF-detection conjecture. The knot meridian is CTF-detected. This conjecture is motivated by the L-space conjecture and the relationship between CTF and NLS slope detection. It remains open; the paper proves the corresponding statement for non-simple $(1,1)$-knots in $S^3$ or lens spaces.