I haven't read it carefully, but the new paper here by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.

I haven't read it carefully, but the new paper here by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.

EDIT: As the comments pointed out, the paper my original answer pointed to only proved it for tight contact structures. However, in a new paper here Licata-Vértesi prove the complete Giroux correspondence.