I'm going to have to re-watch it since I can't remember exactly, but didn't she go back with Crusch because she was injured? So technically the letter could be from Crusch's camp but also be written by Rem because it'd make it more trustworthy coming from her than one of the royal candidates.
Tbh I don't mind if it does turn out to be true
but I hope it at least makes sense rather than the show including a shocking moment just for the sake of having one.
she went back, but it would have been an official letter between candidates. That would be written by either crusch herself, or her first in command. but fine... lets assume for some reason that Rem wrote it.
Subaru should have known that then, or even if he didn't actively know that he would question why it was blank.
if he thought rem could be missing, he would be broken right now, and we already know he wouldn't forget.
he also wouldn't be taking the blame of the letter on himself.
as
seen in this image.
this show has been pretty good about little details, so it really just doesn't add up.
It's not impossible that the writer just forced something and screwed up stuff at the end, or the animation staff is missing something. But given that A. this is based off the rewrite, and B. the writer is known to be working actively with them to make the anime. it seems unlikely.
that all being said, Rem put up so many death flags, it still seems unlikely that she will survive through the end of the next episode even if she isn't present in the fight, but for now this theory seems as odd as the puck = witch theory that was actively proven false this episode once again