Consider a pure braid. I wonder whether there is a criterion that tells me when it splits?

What I mean here with "split": Let the top and bottom ends of the braid be attached to a disk, such that the bottom end of a given strand has the same coordinates as its bottom end. We can think of the braid running inside a cylinder = disc x interval. (cylinder black in the figure 1, strands red)

How do I know (from a braid word ideally) whether I find a disk whose boundary lies in the boundary of the cylinder, with two segments in the disk that follow the same path in the disk, and two segments running from the top to the (blue in the picture 1) that splits the cylinder into two parts in which each strands of the braid are contained?

This is not equivalent to the braid closure forming a split link.

If anyone has a reference or idea, that would be highly appreciated!

Thank you!