Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of a localization of some polynomial ring over $k$.
My question is: Does there exist a surjective $k$-algebra homomorphism $k[X_1,\ldots, X_n]_P \to R$, for some integer $n\geq 0$ and prime ideal $P $ of $k[X_1,\ldots, X_n]$, such that $\mu(\mathfrak m)=\dim k[X_1,\ldots, X_n]_P $?
(Here, $\mu(\mathfrak m)$ is the minimal number of generators of $\mathfrak m$).
Some obvious thoughts: It is enough to show that there exists integer $n\geq 0$ and prime ideal $P $ of $k[X_1,\ldots, X_n]$ such that $R \cong \dfrac{ k[X_1,\ldots, X_n]_P}{I}$ for some ideal $I$ contained in the square of the maximal ideal of $k[X_1,\ldots, X_n]_P$.