The Generalized Cosmetic Crossing Conjecture for Conway tangles

Let $T$ be a Conway tangle with the specified connectivity and no closed components, and let $\{K_n\}_{n\in\mathbb{Z}}$ be the associated family of knots. Say that $T$ is horizontally split when it admits the corresponding horizontal splitting. Generalized Cosmetic Crossing Conjecture. If $T$ is not horizontally split, then the unoriented knots $\{K_n\}_{n\in\mathbb{Z}}$ are pairwise different. This is the tangle formulation of the generalized cosmetic crossing problem; it is open in general, with the paper establishing asymptotic and other partial results.