I haven't read it carefully, but the new paper [here][1] by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.

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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][2] Licata-Vértesi prove the complete Giroux correspondence.

  [1]: https://arxiv.org/abs/2309.11828
  [2]: http://www.arxiv.org/abs/2408.01079