Conjecture on the maximal-elasticity difference invariant

Let $H$ be a transfer Krull monoid over a finite abelian group $G$ with $|G|>4$. Let $\Delta_{\rho}(H)$ denote the set of all positive integers $d$ such that, for every $k\in\mathbb N$, some set of lengths $L_k\in\mathcal L(H)$ with $\rho(L_k)=\rho(H)$ contains an arbitrarily long arithmetic progression of difference $d$ in the sense specified in the definition of $\Delta_{\rho}(H)$. Maximal-elasticity difference conjecture.

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

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

The conjecture concerns the structure of sets of lengths with maximal possible elasticity for transfer Krull monoids. The preceding discussion notes the exceptional small groups and gives partial results, including infinitely many cases where the relevant equivalent statements hold; the assertion is not resolved in general.