On the vanishing of Tor of the absolute integral closure

Let $R$ be an excellent local domain of positive characteristic with residue field $k$ and let $R^{+}$ be its absolute integral closure. If ${\text{Tor}}_1^R(R^{+},k)$ vanishes, then $R$ is Cohen-Macaulay, normal, F-rational and F-pure. If $R$ has at most an isolated singularity or has dimension at most two, then $R$ is regular.