High-distance neighbors conjecture for bridge splittings

Let $K$ be a knot in a 3-manifold $M$ with a $(g,b)$-splitting, where $g\geq 1$. Let $(g',b',n)$ be a triple of non-negative integers with $g'\geq g$ and $b'\geq b$. High-distance neighbors conjecture. There is a knot $K'\subset M$ obtained from $K$ by a single crossing change such that $K'$ has a $(g',b')$-splitting of distance at least $n$. This would extend the paper's result from genus-zero bridge splittings with $b\geq 3$ to higher-genus bridge splittings; the cases with $b\leq 2$ present the main technical difficulty.