Cayley-graph disorder–diameter asymptotic conjecture

Let $G$ be a finite group, let $S$ be a generating set, let $n=|G|$, and let $\mathfrak{D}(\operatorname{Cay}(G,S))$ and $\mathcal{A}(\operatorname{Cay}(G,S))$ denote the graph diameter and average graph disorder, respectively. Cayley-graph disorder–diameter asymptotic conjecture. Under appropriate conditions, there is a function $\epsilon(\operatorname{Cay}(G,S))=o(\mathcal{A}(\operatorname{Cay}(G,S)))$, determined by the number of generators in $S$ of order $2$, such that

$$n\,\mathfrak{D}(\operatorname{Cay}(G,S))=\mathcal{A}(\operatorname{Cay}(G,S))-\epsilon(\operatorname{Cay}(G,S)).$$

The conjecture refines the observed inequality $n\mathfrak{D}(\operatorname{Cay}(G,S))\leq\mathcal{A}(\operatorname{Cay}(G,S))$ and proposes that the discrepancy is asymptotically negligible apart from its dependence on involutory generators. The appropriate conditions are not specified in the supplied text.