The Cosmetic Crossing Conjecture for Conway tangles

Let $T$ be a Conway tangle, and let $K_0$ and $K_1$ be the knots obtained from the associated family of tangle fillings. Say that $T$ is horizontally split when it admits the corresponding horizontal splitting. Cosmetic Crossing Conjecture. If $T$ is not horizontally split, then $K_0$ and $K_1$ are different as unoriented knots. This is a tangle reformulation of the Cosmetic Crossing Conjecture; the paper proves related special cases and asymptotic consequences, while the general conjecture remains open.