Grassmann-odd three-point functions of conserved supercurrents in 3D $\mathcal{N}=1$ SCFT

We consider the analytic construction of three-point functions of conserved higher-spin supercurrents in three-dimensional $\mathcal{N}=1$ superconformal field theory which are Grassmann-odd in superspace. In particular, these include the three-point functions of the supercurrent and flavour currents, which contain the three-point functions of the energy-momentum tensor and conserved vector currents at the component level. We present an analytic proof for arbitrary superspins that these correlators do not possess a parity-violating contribution. We also prove that the parity-even contribution is unique, and exists (under an assumption that is well supported by the computational approach of arXiv:2302.00593) for arbitrary superspins. The construction of the parity-even sector is shown to reduce to solving a system of linear homogeneous equations with a tri-diagonal matrix of co-rank one, which we solve explicitly for arbitrary superspins.