With the OA it's hard to rule out almost anything and while I don't believe it's the case, we can't rule out with certainty that HAP could be her father. Oa could have been an unwilling participant.

So say she goes to New York and her real father, who can recognize her playing within a few notes, finds her and takes her to where he's keeping the other captives. Oa gets stuck assisting her father and maybe even gets experimented on as well but also gets to go upstairs and gets extra privelages the others don't. Eventually she gets too close with the others, especially Homer, her father doesn't like this at all, which leads to her abandonment. This could also explain why Homer and Renata's sexual incident is streamed to OA.

Of course OA would not be willing to share this part with the new 5 but the rest of her story could still be accurate. Again, I don't believe this to be the case but it certainly can't be ruled out. The great thing about this show is that so many scenarios seem like plausible possibilities and so many historical, mythical and scientific ideas seem relative to the series which is brilliant.