Strong 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 $D(G^n)$ denote the diameter with respect to a generating set, using words in that set. Write ${\mathrm{rank}}(G)$ for the minimum size of a generating set of $G$. Strong diameter conjecture. The diameter $D(G^n)$ is at most

$$n\bigl(|G|-{\mathrm{rank}}(G)\bigr).$$

The bound would give a linear, rather than exponential, estimate in $n$ for direct powers; it is proposed as a stronger form of the accompanying assertion concerning a suitable minimum generating set.