On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.