Because I feel the contents speak for itself. Kimblee-san has the most to lose from that note existing, whereas Fulbright-san stands only to gain from it for no obvious reason. Fulbright-san obviously wants us to kill Kimblee-san, and he's the only one strongly pushing for us to do so. When asked about the note, however, he can't defend himself - all he can say is "I don't know" and "it is what it is," and if anything he's provided us with a good reason to suspect him given that he's pointed out that he can really only know his own alliance.