We literally follow Mustaţă’s proof with two modifications. The proof requires Mumford’s theorem on Castelnuovo-Mumford regularity for the projective scheme $X$ over $R$ which holds also in this more general setting [BS13, 20.4.13]. Furthermore we replace the use of Fujita’s vanishing theorem to the sheaves $\mathscr{F}_{j}:=\mathcal{O}_{\mathscr{X}}(K_{X/k}+T_{j})$, $j=1,\dots,r$ and the ample divisor $(d-i)H$ by an application of Keeler’s generalization [Kee03, Thm. 1.5]. ∎