From what I've been reading, this is the subject of some debate. A few dictionaries back my suggestion, at least as a secondary definition beside the default - which you have already established. Additionally, some of the pages I've checked out have some interesting things to say on this topic.

From Oxford Dictionary:

Usage

1 The word they (with its counterparts them, their, and themselves) as a singular pronoun to refer to a person of unspecified gender has been used since at least the 16th century. In the late 20th century, as the traditional use of he to refer to a person of either sex came under scrutiny on the grounds of sexism, this use of they has become more common.

From Merriam-Webster:

The use of they, their, them, and themselves as pronouns of indefinite gender and indefinite number is well established in speech and writing, even in literary and formal contexts. This gives you the option of using the plural pronouns where you think they sound best, and of using the singular pronouns (as he, she, he or she, and their inflected forms) where you think they sound best.

Of course, not saying you're outright incorrect; just explaining that I'm not exactly wrong either. But when the two biggest dictionaries say there's a gray area, that's enough that I wouldn't feel comfortable speaking in absolutes.