An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Question

Find a counter example to the following statement:

Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the canonical inverse image in $G$ of the center of $G/A$, then $$Z(Z_G(A)) \subset R_{p-1}(G)$$ where $R_{p-1}(G)$ is the set of right $(p-1)$-Engel elements in $G$.

Thanks in advance.