Not exactly, Fi said she would have a sleep without end, whether that's the same as dying, we can't be sure. I recall another article on here that suggested that Fi's bias towards her original master (SS Link) influenced her greatly, and this is why she initially rejected OoT Link because he was...
Yeah, I spent a couple of days writing this and while I am prepared for some other fans to find logical contradictions, I feel confident that all incarnations of Link are SS Link reincarnate, at least in the examples I provided.
So, not sure if it'd be considered spoilers but to anyone who hasn't played Skyward Sword, then spoilers, I guess?
Also, fair warning.... this is a 10-page document coming your way down below, so if you're not much of a reader then feel free to read at your own leisure.
An Analysis of The...