The characterization of groups with singleton maximal-elasticity difference set

Let $G$ be a finite abelian group with $|G|>4$. Write $\rho(G)$ for its elasticity and let $\Delta_{\rho}(G)$ be the set of all $d\in\mathbb N$ such that, for every $k\in\mathbb N$, there \exists a length set $L_k\in\mathcal L(G)$ that is an almost arithmetic progression with difference $d$, length at least $k$, and $\rho(L_k)=\rho(G)$. The maximal-elasticity difference-set conjecture. Then

$$\Delta_{\rho}(G)=\{1\}$$

if and only if $G$ is neither cyclic nor an elementary $2$-group. This conjecture seeks a precise group-theoretic characterization of when maximal-elasticity length sets have only difference $1$; the supplied text does not state whether it has been proved or disproved.