The answer their is definitely correct (no matter what anybody else tells you). But the answer her is also correct.
To see why their is correct, amd maybe why it's preferable, consider the related sentence:
Neither Jane nor John remembered to bring ______ camera.
You can't use either his or her in this sentence — if you use his, you're specifying that the camera was John's and if you use her, you're specifying that it was Jane's. You could use his or her, or maybe her or his, but their is much better.
In the original sentence, you can also use her because both of the subjects are female.