The complete regular ring in Cohen's structure theorem can be chosen to have dimension equal to the embedding dimension of the starting ring?

Question

Let $(R, \mathfrak m)$ be a Noetherian complete local ring. Then by Cohen structure theorem, we have that $R$ is a homomorphic image of a complete regular local ring $(S, \mathfrak n)$ (https://stacks.math.columbia.edu/tag/0323).

My question is: In Cohen structure Theorem, can we always choose a complete regular local ring $(S, \mathfrak n)$ such that $\mu(\mathfrak m)=\dim S $ ?