Cayley-graph diameter and minimal-generator conjecture

Let $G$ be a finite group, let $S$ be a generating set, let $n$ denote the group order, let $g_{\mathrm{min}}$ be the minimal number of generators of $G$, and let $\mathfrak{D}(\operatorname{Cay}(G,S))$ denote the graph diameter. Cayley-graph diameter and minimal-generator conjecture. If

$$\mathfrak{D}(\operatorname{Cay}(G,S))\geq \frac{n}{g_{\mathrm{min}}},$$

then $G=(\mathbb{Z}/2\mathbb{Z})^k$ and $g=k$ for $k=2,3,4$. Moreover, the value $\mathfrak{D}(\operatorname{Cay}(G,S))\,g_{\mathrm{min}}/n$ is directly related to some group property. The conjecture is based on computations of small finite groups, where the corresponding upper bound had only the stated exceptions and equality occurred only for the listed elementary abelian $2$-groups. The group property governing the normalized value is not identified in the source.