Classification of charge-conserving loop braid representations

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category ${\mathsf{M}\mathsf{a}\mathsf{t}\mathsf{c}\mathsf{h}}^N$ of charge-conserving matrices (specifically ${\mathsf{M}\mathsf{a}\mathsf{t}\mathsf{c}\mathsf{h}}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{M}\mathsf{a}\mathsf{t}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that representations fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree $N$.