If you look from a RD point of view, Crono is dead.
In RD, next to the end of scenario1, if you ask Kid "Why are we after the Frozen Flame?"
She'll say:
"I made a promise with my big sis... to get the Frozen Flame, and bury it for her...
[...]At her old friend's grave...
[...]I was still little at the time, so I can't remember much of it... But there was this friend of hers who she'd grown
up with, who died a while back. She said she wanted him to be able to rest with the Frozen Flame..."
I think that she is talking about Crono anyway~
EDIT: just to add, since I don't think that the fall of Guardia was planned back when RD was made, I think the ending where Crono dies in CT is the real one. Reviving him was just a side quets anyway~