The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need the case $ \Delta =0$ in which case, weak log terminal might be the same as just terminal.). Does anyone know how to prove this, or a place where this is proved?