How to fix a broken flap FCC connector flip-lock ZIF
I have worked a lot with FPC connectors, and that pin layout looks very much like is used by some of the Hirose and Panasonic FPC connectors, however, their websites don't show as having an 84 contact connector in production.

In my experience, I have found that when it comes to mass produced electronic goods, some of the connectors are custom made, and therefore near to impossible to find as a replacement.

I assume that the likely hood of working on this connector in the future is fairly low, and therefore I suggest that you insert the FPC with a spacer, and then hold it in place with some hot glue, or any kind of adhesive, that can be easily removed, in case you needed to do some maintenance on the projector.