Weak diameter conjecture for direct powers of finite groups

Let $G$ be a finite group, let $G^n$ denote its $n$-th direct power, and let $A$ be a generating set of minimum size for $G^n$. Write ${\mathrm{rank}}(G)$ for the minimum size of a generating set of $G$, and let ${\mathrm{diam}}(G^n,A)$ be the diameter with respect to $A$. Weak diameter conjecture. There exists a generating set $A$ for $G^n$ of minimum size such that

$${\mathrm{diam}}(G^n,A)\leq n\bigl(|G|-{\mathrm{rank}}(G)\bigr).$$

This is the weaker companion to the assertion for the diameter with respect to any generating set, and is intended to provide a polynomial bound for the diameter of direct powers.