Is it correct to say "The list of reports a user will see may depend on the permissions group(s) they are in."?
I have asked the writer to change the text to say "users" to match "they", but he is resisting.
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Depends on the formality of the piece. In actuality, this use has been acceptable for a long time in formal and informal writing, but sometimes a less-informed superior may not be aware of this. In that case I suggest the following guideline:
Informally, "they" is an acceptable gender-neutral third person singular pronoun. Formally, "they" would not be acceptable here as it is exclusively for third person plural prounoun use, and it should be reworded somehow--your suggestion would be the easiest.
(Edited to take into account both the actual legality of singular they as well as the possibility of superiors unaware of this)