Yes – my experiments back this up. In practice, this means that if I have (say) a “staff” group with flair, and add staff to another “primary” group which also has flair, I have to then toggle the “staff” group off primary, save, then back onto primary for them to continue showing up with the “staff” flair.
It’s a bit frustrating, but it only affects my use of flair a few times a year, so it’s something I just live with for now. 