Centralizer-count conjecture for groups with quotient $\mathbb{Z}_{p^n}\rtimes\mathbb{Z}_{p^n}$

Let $p$ be a prime number, let $n$ be a positive integer, and let $G$ be a group with center $Z$ such that

$$G/Z\cong \mathbb{Z}_{p^n}\rtimes \mathbb{Z}_{p^n}.$$

Here, $\operatorname{Cent}(G)$ denotes the set of distinct centralizers of elements of $G$.

Centralizer-count conjecture. The number of distinct centralizers of $G$ satisfies

$$|\operatorname{Cent}(G)|=(p+1)^n+1.$$

The conjecture generalizes the preceding calculation for the case corresponding to $n=2$, where the number of distinct centralizers is $(p+1)^2+1$. Its validity for arbitrary positive integers $n$ is not established in the supplied text.