There is no "test of cure" for HPV the way there is for some other STI's, like chlamydia for example, where you can prove you no longer have it. HPV can exist in very small numbers in deeper skin cells, and there is no way to know for certain if 1) it's no longer able to be transmitted at a given point or 2) if it might "wake up" in future.
However....this much is clear. For the vast majority, the immune system dominates HPV at some point (usually 6-24 months) and having long-term issues with recurrences, that is, warts or cervical cell changes coming back over and over, year after year, is not at all common. Given that most women who test "positive" on HPV DNA tests in research studies eventually test "negative" it SEEMS the infection is either suppressed/cleared effectively, leading many (perhaps fair to say most) researchers to believe it's unlikley the virus will always remain in a state where it can be transmitted.
But how to prove that for sure? Well, no one has so far and while the design of research studies is not my area of expertise, transmission studies between couples might be difficult, expensive, and take a very long time to do.
As I read this post I see qualifiers like "seems," "probably," "perhaps," and so on - I don't like having to dance around with my answers and wish I could just say "yep" or "nope" and answer you directly. The lack of certainty is frustrating for all of us, to be sure. I hope this offers some insight though, but let's keep discussing it if not!
All the best,