On a relation between the self-linking number and the braid index of closed braids in open books

We prove a generalization of the Jones-Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several examples that do not satisfy a naive generalization of the Jones-Kawamuro conjecture.