Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $R$?
Remark 1: It suffices to show that $S^+/p$ is flat over $R/p$, but I don't know how to see this.
Remark 2: I am mostly interested in the situation when $R$ is non-noetherian by itself, but $R[1/p]$ is. But I don't know whether this result holds even in the noetherian case.
If it helps, feel free to add extra conditions on a pair $(R, R[1/p])$. For example, in my main case of interest $R$ is $p$-adically complete and $R[1/p]$ is regular. Though I am not sure how useful it is.