Relative Giroux correspondence for contact manifolds with convex boundary

Let $M$ be a compact contact manifold with convex boundary. A contact partial open book is a triple $(S,W,\phi)$, and let $M_{(S,W,\phi)}$ denote the contact manifold associated with it.

Relative Giroux correspondence. Every compact contact manifold with convex boundary is contactomorphic to

$$M\cong M_{(S,W,\phi)}$$

for some contact partial open book $(S,W,\phi)$.

This is the relative analogue of the Giroux correspondence, extending the open-book description of closed contact manifolds to contact manifolds with convex boundary. The source states it as a conjecture, and no resolution is given here.