Well, my POV.
The naysayers here are saying. "The house is crooked! And I never liked the color, anyway. Tear it all down, and start from scratch". The house, here, is an analogy for the theory of GR, which has been built up painstakingly over time. Not liking the color is an analogy for the apparent dislike of the theory by some for reasons that don't have much to do with how well it works.
The rest of us (and I'd put myself in this group are saying). "The house is basically structually sound, living in it is quite comfortable".
We're talking about an error of a few mm/sec here of objects moving at kilometers/second!
Certainly it's something that should be looked at, but the natural scientific process of improving our knowledge of the Earth's actual gravity field, including all the multipole-moments due to its not-quite-spherical shape is likely to resolve the issue. We can't rule out more interesting explanations at this point (for instance, a dark matter halo), but the data isn't good enough to really strongly suggest anything in particular to improve our predictions.
Meanwhile, the actual work continues, quietly, as more and more measurements are made. I'm not sure of the details of the work, but it appears we have a WGS-84 from 1984, and EGM from 84, 96, and 2008.
Conspiracy theoreists may delight in the fact that some of this ungoing work is apparently being done by the government - through the NGA - the national geospatial intelligence agency. A mysterious, spooky-sounding name, to be sure, it sounds like fertile ground for all sorts of fantastic speculations. Which I shall prudently omit on the grounds that someone would probably take me seriously.