Is $\dim (M/xM) = \dim M - 1$ for some $x \in m$ implies $x$ is an $M$-regular element ? where $m$ is the maximal ideal of a local ring $R$.

Let $R$ be a Noetherian local ring with maximal ideal $m$, and let $M$ be a finitely generated $R$-module. It is known that if $x \in m$ is an $M$-regular element, then the dimension of $M/xM$ is equal to the dimension of $M$ minus 1, i.e., $\dim (M/xM) = \dim M - 1$.

My question is whether the converse of that statement is true, i.e.,

If $\dim (M/xM) = \dim M - 1$ for some $x \in m$, then $x$ is an $M$-regular element?

It is known that if $M$ is a Cohen-Macaulay module, then the converse statement is true. However, the question remains whether this result holds in general for all finitely generated modules over a Noetherian local ring.