There were traditionally two ways to handle unknown gender references. Once was just to assume it was a man, and the other was to use two pronouns, such as 'his or her'.
No. The traditional way, going back to Jane Austen and Oscar Wilde and even Shakespeare, was to use "their."
In the 1800s, it became "he."
By the end of the 1900s, we were back to "their" in many cases, and now almost universally again.
I have done a lot of research on this. The people taught English in the 1950s-1970s think the change to "their" is some feminist initiative. It's really our language going to back to what it was before the absurd "he" for "anyone."
Can you imagine addressing one man and one woman and saying "I can't say who yet, but one of you is going to lose his job as a result of the merger"? Suddenly "his" for either sex makes less sense, doesn't it?
HOWEVER - in this case, it was one person, and presumably that person is KNOWN, so there is no reason to not use either "his" or "her" depending on the actual gender of the person whose job was lost. (Unless you wanted to protect that person's identity and thus minimize the amount of personal information about the now-jobless person so people didn't start guessing who it was.)