Well-defined 3D TQFT conjecture for the assignments of the partition function

Let $Z$ be the assignment constructed from the stated values on half-planes and elementary braid discs, with the monoidal structure given by convolution. In particular, the elementary braid assignments are $\mathsf{Z}^\bullet(S_{\sigma_k^{\pm1}})=\mathcal{C}_\pm^{(k)}$, and the tautological half-plane is assigned $\mathcal{C}_1\otimes\Lambda^\bullet\mathcal{B}$.

Well-defined 3D TQFT conjecture. The above assignments of the values of $Z$ are part of the data of a well-defined 3D TQFT.

This would promote the constructed braid and defect assignments to a genuine three-dimensional topological quantum field theory. The source explicitly leaves the statement as a conjecture and says that a proof is forthcoming.