On half-factoriality of transfer Krull monoids

Let $H$ be a transfer Krull monoid over a subset $G_0$ of an abelian group $G$ with finite exponent. Then every non-unit $a\in H$ can be written as a finite product of atoms, say $a=u_1\cdot \dots \cdot u_k$. The set $\mathsf L(a)$ofallpossiblefactorizationlengths$k$iscalledthesetoflengthsof$a$,and$H$issaidtobehalf-factorialif$|\mathsf L(a)|=1$forall$a\in H$\mathrm{.}Weshowthat,if$a \in H$and$|\mathsf L(a^{\lfloor (3\exp(G) - 3)/2 \rfloor})| = 1$,thenthesmallestdivisor-closedsubmonoidof$H$containing$a$ishalf-factorial\mathrm{.}Inaddition,weprovethat,if$G_0$isfiniteand$|\mathsf L(\prod_{g\in G_0}g^{2$\mathsf{o}\mathsf{r}\mathsf{d}$(g)})|=1$,then$H$ is half-factorial.