Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \subset X$ containing the $x_i$?

To give some context, Lemma 4 of section 6.4 of "Neron models" of Bosch, Lutkebohmert, and Raynaud says that the answer should be 'yes' but I don't understand its proof (how does EGA IV, 8.10.5 apply to the $u_j$ constructed there?).