Unfortunately the ‘white’ LEDs with high CRI ratings fare poorly. Partially because the CRI measurement system is badly flawed, resulting in poor reproduciton in skin-tones specifically. Hopefully the industry will switch to CQS which gives a much more true measure.
I see some manufacturers out there have been supplementing with R9 and R13 values over the last couple of years, but that seems like a patch to me. Since I only buy lamps for household (non-critical use) adding more skin tone colors into the rating system is probably a plus at the increase of more things to compare. CQS rating is not something I have come across until today, but it sounds like a much better system overall. I would also imagine (hope?) that the ratings are generally better for component parts than consumer products.