Bode's braided open-book conjecture for fibered links

Let $L$ be a fibered link in $S^3$. A braided open book is an open book in $S^3$ whose binding is braided relative to the standard open book with unknotted binding; equivalently, its binding is the closure of a $P$-fibered braid, a braid whose argument map induces a fibration. Bode's braided open-book conjecture. Every fibered link in $S^3$ is the binding of a braided open book, or equivalently, the closure of a $P$-fibered braid. The conjecture is known for fibered links of braid index at most $3$ and for closures of homogeneous braids, while the general case remains open.