On the structure of length sets with maximal elasticity

Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor. Then every non-unit $a\in H$ has a factorization into atoms, say $a=u_1\cdot \dots \cdot u_k$ where $k$ is the factorization length and $u_1,\dots ,u_k$ are atoms of $H$. The set $\mathsf{L}(a)$ of all possible factorizaton lengths is the length set of $a$, and $\rho (H)=\sup \{\max \mathsf{L}(a)\mathrm{/}\min \mathsf{L}(a)\text{\hspace{0.17em}⁣}:a\in H\}$ is the elasticity of $H$. We study the structure of length sets of elements with maximal elasticity and show that, in general, these length sets are intervals.