Binding conjecture for open books with prescribed negative twists

Let $\mathcal M_k$, for $k\geq 0$, be the set of closed, oriented, smooth $3$-manifolds admitting open book decompositions whose monodromies contain at most $k$ negative Dehn twists, and set $\mathcal M_{-1}=\emptyset$. Let $M\in\mathcal M_k\setminus\mathcal M_{k-1}$, and let $L$ be any link in $M$. Binding conjecture. There exists a knot $L'$ such that $L\cup L'$ is the binding of an open book decomposition of $M$ whose monodromy contains exactly $k$ negative Dehn twists. This proposes that the binding of an open book can be chosen with substantial freedom even when the number of negative Dehn twists required by the manifold is prescribed; the source provides no resolution of the conjecture.