in the books, the letter was likely forged by mance rayder, so its possible the show is following that theory
however, in the books, mance rayder is still alive, and in the show he is not. the show very much disvalued mance's character, by removing a lot of his crew and his child from the story. so as far as the show is concerned, it probably isnt mance, but who it really is is hard to say.
i only disagree with the baelish theory because baelish is probably smart enough not to send letters under different names using the exact same wax. sure, the show has been showing wax being pressed for a long time, but i think its to fulfill some weird fetish among the cast and crew than some elaborate setup for some pretty minuscule result, the kind most people wouldnt notice.
but hey, the writers dont even fucking care anymore. even in theyre interviews for the show theyre just not as interested as they were in the earlier seasons.