It is extensively used and cited the following statement due to Giroux:
Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and open book decompositions of $M$ up to positive stabilization.
Given such a contact structure, the existence of an open book supporting the contact structure is proven, for example, in Giroux's Géométrie de contact: de la dimension trois vers les dimensions supérieures.
I am asking for a complete proof of the uniqueness part of the result. At this point I would be surprised if somebody provided me with a link to a peer-reviewed paper containing a proof of the result (which is funny because this is widely acknowledged as a "theorem" inside and outside the field of contact geometry). Usual citations include the above paper (which does not contain a proof of the statement), some book that has been "in preparation" for years or even "transparencies from a seminar"!
So, links to detailed lecture notes or a proof itself will be appreciated. I know for example the existence of these Lectures on open book decompositions and contact structures by J. Etnyre, but they are somehow sketchy to my taste. I am not an expert in the field and I can't complete all the exercises left to the reader or fill in all the gaps in the "sketches of a proof".