We know that a finitely generated $R$-module $M$ satisfies the $(S_n)$ condition if $$\operatorname{depth} M_p \geq \min(n,\dim M_p)$$ for every $p\in \operatorname{Spec}R$. It's well known that Cohen-Macaulay rings satisfy $(S_n)$ for all $n \geq 0$. Now is the following conclusion true:

If $A$ is a quotient of a Cohen-Macaulay local ring and satisfies $(S_n)$ then the completion $\hat{A}$ also satisfies $(S_n)$?

I want to use the proposition 2.1.16 from *Cohen-Macaulay Rings*, Bruns-Herzog.

Commutative ring theory, page 185, Exercise 23.2. $\endgroup$1more comment