When the Nazgûl lose their bodily form, what happens to their rings?

[Galin]

Originally Posted by Morthoron

I don't think Tolkien reconciled the nine rings conclusively. It's one of those plot holes pitting the fabric of such a sprawling tale. And the inconsistency in the plot logic is nowhere more noticeable than after Isildur cuts the One Ring off Sauron's finger. If Sauron had the Nine, then they would have been lost in the fall of Barad-Dur (which would not be raised again for centuries).

But why can't Sauron have taken the Nine back in the Third Age? That appears to be the indication from Of The Rings Of Power And The Third Age, as I read it anyway.