  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 $\mathsf{Match}^N$ of
charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same
rank-$N$ charge-conserving monoidal subcategory of the monoidal category
$\mathsf{Mat}$ 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 $N$.
