The Baltimore Rock Opera Society fired its recently hired executive director, allegedly because she was wearing her hair in cornrows and thus engaging in supposed cultural appropriation. The concern wasn't that cornrows were seen as unprofessional as such, but rather that a white woman was wearing a hairstyle that is seen as culturally black. (Katie Herzog at Blocked and Reported has a very good new podcast episode on this.)
Such a firing, I think, is illegal under federal antidiscrimination law, though it's possible (but not certain) that BROS may have a First Amendment defense, given the particular job that the woman held.
[1.] The gender identity discrimination precedent: The key precedent here is Harris Funeral Homes v. Stephens, the trans companion case to Bostock v. Clayton County. In those cases the Court held that gender identity and sexual orientation discrimination is sex discrimination, and therefore banned by the Title VII prohibition on sex discrimination. Here's the key part of the logic behind that argument: