The eventual interval conjecture for unions of sets of lengths

Let $H$ be a Krull monoid as in Theorem 5.3, with associated set $G_P$, and suppose in addition that

$$G_P=G_r^+\cup -G_r^+.$$

For $k\in\mathbb N$, let $\mathcal U_k(H)$ denote the union of the sets of lengths of elements of $H$ having a factorization of length $k$. The eventual interval conjecture. There exists $k^*\in\mathbb N$ such that, for each $k\geq k^*$, $\mathcal U_k(H)$ is an interval. This conjecture strengthens the first assertion of Theorem 5.3 by asserting that the additive constant can be taken to be $M=0$ for all sufficiently large $k$. Its resolution is not supplied here and remains open.