Molecularity of the class $\mathcal{C}_3$

Let $\mathcal{C}_3$ be the class of additive submonoids of $\mathbb{Q}_{\ge 0}$ designated by the source, and call a monoid molecular if every non-invertible element is a molecule, that is, has exactly one factorization into atoms.

Molecularity conjecture for $\mathcal{C}_3$. The class $\mathcal{C}_3$ is molecular.

The conjecture is proposed in the direction of earlier results establishing molecularity for the classes $\mathcal{C}_1$ and $\mathcal{C}_2$. The supplied text does not provide a resolution, and the precise definition of $\mathcal{C}_3$ is not included in the candidate context.