Both are correct. "Shy away" would insinuate that they are staying away because they are nervous, whereas "stay away" is more of a statement of fact about what they are doing. For your example, because it the topic is being avoided because it hard to talk about and you would be nervous to talk about it, both will be correct. But in another example, if someone is avoiding someone because they are angry with them, you would say that they are "staying away" (because they are physically avoiding them) but not "shying away" (it is not because of nerves).

Both "shy (away)" and "stay away" are correct in this sentence. As a verb "shy" is considered more sophisticated, high talk or poetic. (It's fancy). It is not common but not rare either. "stay away" is more normaly undersood. Example: "Praying mantises usually stays away from these trees." "The praying mantis kind commonly shies away from such trees." Same meaning, different way of saying it.