Global $+$-regularity of regular del Pezzo surfaces in mixed characteristic

Let $R=W(k)$ be the ring of Witt vectors over an algebraically closed field $k$ of characteristic $p>2$. Let $M$ be a three-dimensional regular integral flat projective $R$-scheme such that $H^0(M,{\mathcal{O}}_M)=R$ and the anticanonical sheaf ${\omega }_M^{-1}$ is ample. We show that $M$ is globally $+$-regular if the closed fiber $M_k$ is reduced.