I see that the latest action at
https://github.com/haskell/directory/pull/136 is only two days ago, so hopefully you and Phil will come to a conclusion soon.
Unfortunately, CLC does not know release schedule of GHC 9.6 either. This sits squarely within GHC developers, please ask them for GHC 9.6 feature freeze date. From the previous experience, it usually occurs pretty soon after the previous release, maybe within a month or two, so probably September-October 2022, but it’s better to ask directly.
The process of updating is pretty straightforward: make a Hackage release, mirror it as a git tag, raise an MR to bump GHC submodule to that tag.
Best regards,
Andrew