The use of their as a singular pronoun has a long history in English - at least as long ago as the late 13th century. Historically, it has been used as an indefinite antecedent (example: "Someone left their umbrella"). It modern times it has become charged with politically motivated energy, in that some groups have chosen it as a battleground in the "gender" war. The American Association for Psychological Science, that well-known authority on English grammar, has decided that their (used as a specific singular), along with ze, hen and other pronouns, should be accepted into the corpus of English vocabulary. Not everybody agrees.
Perhaps we should bear in mind that, strictly, gender (in its primary meaning) has to do with grammatical classification, and what we are really talking about in the current "gender" debate is sex or biological classification.
Until relatively recently the word ethnic meant heathen, non-Christian, and gay could only mean happy. Both these words have made a transition to new meanings that have superseded the old. "Their" used as a singular specific has not quite managed that trick yet. Although it looks set to do so, with English you can never be sure.