We (York Caving Club, with members from NYMCC/RRCPC) are working mostly up on The Stang between Reeth and Barnard Castle, focusing on the caves around Jinglepot Hole and Seavy Sikes Pot. Here, the maze cave (Draughting Hole/Missed Pot) keeps on growing and the system is approaching the 9-10km mark with many walking height open passages (and many smaller ones) still unexplored.
We've done a bit at the sinks around Doorgill Head too, and we have an ongoing project there (albeit ongoing slowly).
As for the sinks and shakeholes higher up on Faggergill Moor, I think some of our team may have looked at these but not extensively. They are on access land which is great, although we'd encourage anyone planning on heading up there for more than a recce to establish some favourable relations with the local gamekeepers particularly as the area is used for shooting.
Obviously, one potential discovery could be a route into the natural caverns of Faggergill, which are currently only accessible via a very long trip through Faggergill Mine. Access there is highly restricted because, I think, the mine contains vulnerable mining heritage, so if you did get into Faggergill then you would need to engage with the local mine exploration community.
The alternative could be the discovery of the fabled Roughton Keld master cave... something we are hoping to find from our end, although it's not totally clear at the moment whether the water sinking around Doorgill Head and Jinglepot Hole goes to Roughton Keld or Eller Beck.
Some of our team have given talks on these things at the last few Hidden Earths.
I know that others on this forum are working in the area from a different angle, and they may wish to comment too (I'm not sure how much they are wanting to discuss/disclose).
This is a pretty bleak place, requiring very long walk-ins from the road, and extremely exposed to the prevailing winds and rain. Even those large and obvious shakeholes are likely to require an extensive effort involving many sessions and lots of equipment. Plus, it requires careful efforts to build and maintain good gamekeeper/landowner relations. It's certainly not an easy area to explore!
However, the whole area probably has a lot of potential for anyone wanting to put in the effort.