On the Cohen-Macaulay property of the Rees algebra of the module of differentials

Let $R$ be an algebra essentially of finite type over a field $k$ and let ${\mathrm{\Omega }}_k(R)$ be its module of K"ahler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)=0$, we prove that ${\mathrm{\Omega }}_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of $R_{\mathfrak{p}}$ is at most twice its dimension.