Is Ryuk required to write Misa's name in his Death Note at the end of her life? Death Note Rule LXIV states:


  gods of death are obliged to confirm the death of the first owner and
  write down that humans name in his/her Death Note


Misa was the first human owner of the Death Note that was brought into the human world by Rem. Misa then gave up this Death Note, and then picked it up again later, and this time the Shinigami owner was Ryuk (Light made him and Rem switch notebooks). Obviously the responsibility of killing Misa would normally be Rem's, since she was the first to "drop" that DN in the human world. But then Ryuk also "dropped" it after he got ownership of it, and Misa "picked it up" and became the owner for the second time. Rem is dead, so she can't kill Misa. Does this mean that the responsibility transfers to Ryuk? 

I know that the anime makes it look like Misa was about to commit suicide. And it looks like it's her idea, too--I seriously doubt Ryuk would have made her go through all that trouble (wearing the "handmaiden of death" outfit and fancy makeup, traveling to the place where Light died,etc). But we don't actually see her die. Would Ryuk be the one to kill her, now that her life is about to end due to her own intentions? 

Does anyone know of anything in the manga, creator commentary, etc., that might clear this up?

(Rule is from the wiki here http://deathnote.wikia.com/wiki/Rules_of_the_Death_Note)
That rule is an expansion on the rules which state cases when Gods of Death are allowed to return to their own world (more specifically it's the second part of "How to Use: LXIV")

Point one of those rules start with:


  The following situations are the cases where a god of death that has brought the Death Note into the human world is allowed to return to the world of gods of death


While it is true that at one point Ryuk had ownership over Misa's Death Note he wasn't the one who brought it to the human world, so in Misa's case those rules don't apply to him.