Split booklink isotopy conjecture for open book 3-manifolds

Let $\lambda$ be a booklink in an open book 3-manifold representing a split link. A split booklink is a booklink admitting a splitting sphere intersecting each component of the binding in two points. Split booklink isotopy conjecture. There is a finite sequence of booklink isotopies and exchange moves which converts $\lambda$ into a split booklink. This would extend the split-booklink result from the setting considered in the paper to arbitrary open book 3-manifolds; the source does not provide evidence that the conjecture is resolved.