Looks like beating Grand just got a lot harder. Minor FF6 Advance spoiler.

But if the Melody doesn't wear off, he's still mortal. Therefore, subject to death by old age.

Also, counterpoint: Comic #471 shows that Eternion at least has some way of crossing "dimension sets", and Comic #418 says that the Grand has lesser versions of each Eternal under him. If Eternion had the power to cross dimensions, so should Maxima.

Counter-counterpoint: Comic #419, where they discuss Undimensioning Maxima, says it sends the target into "dimensional stasis", not another dimension. Also, Comic #886 shows that Eternion didn't reveal all his powers to the council that granted Maxima his extra powers. Any power to jump unaided between dimensions would likely have been one of the powers he hid.