Relation between $\operatorname{depth}(R)$ and $\operatorname{depth}(R/xR)$.

Question

Let $(S,\mathfrak m)$ be a local regular ring, $I\subset \mathfrak m$ an ideal and $x\in \mathfrak m\setminus I$.

If $x$ is regular on $R=S/I$, then $\operatorname{depth}(R/xR)=\operatorname{depth}(R)-1$ and $\dim(R/xR)=\dim(R)-1$.

Also, without any conditions on $x$, we have $\dim(R/xR)\geq \dim(R)-1$.

My question is the following: In general, is it true that $\operatorname{depth}(R/xR)\geq \operatorname{depth}(R)-1$?