The characterization conjecture for Krull monoids by systems of sets of lengths

Let $M$ and $M'$ be Krull monoids with respective finite abelian class groups $G$ and $G'$, and suppose that each class of both groups contains at least one prime divisor. Assume also that $\mathsf{D}(G)\geq 4$.

Characterization conjecture. If

$$\mathcal{L}(M)=\mathcal{L}(M'),$$

then $M\cong M'$.

This is the Characterization Problem in factorization theory: whether systems of sets of lengths determine Krull monoids under these hypotheses. The source states that the conjecture remains open, although it is known under certain additional conditions.