The conjecture on maximal elasticity differences in transfer Krull monoids

Let $G$ be a finite abelian group, and let $H$ be a transfer Krull monoid over $G$, meaning that there is a transfer homomorphism $H\to\mathcal B(G)$ to the monoid of zero-sum sequences over $G$. Let $\Delta_{\rho}(H)$ denote the set of positive integers $d$ such that, for every $k\in\mathbb N$, some set of lengths $L_k\in\mathcal L(H)$ with maximal elasticity has a sufficiently long arithmetic progression of difference $d$ in the prescribed structure. The maximal elasticity difference conjecture. If $|G|>4$, then

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

if and only if $G$ is neither cyclic nor an elementary $2$-group.

This conjecture concerns the fine structure of sets of lengths having maximal elasticity and was first formulated in the cited earlier work. The supplied text does not state whether it has been resolved.