Let $X$ be an affine variety such that $X-p$ is Cohen-Macaulay, and let $\pi \colon \widetilde{X} \rightarrow X$ be a standard blow-up of $X$ with respect to $\mathcal{O}_X$, centered at $p$. It is known that $\widetilde{X}$ is Cohen-Macaulay.
If the blow-up of $X$ at the maximal ideal corresponding to $p$ is Cohen-Macayulay, is it true that such blow-up is standard?
