The quantum $SO_3$ spin-Chern-Simons correspondence with BM spin TQFTs

The quantum $SO_3$ spin-Chern-Simons theory assigns invariants and state spaces to spin manifolds. For an odd $SO_3$ level of the form $2m-1$, and for the BM spin TQFT at level $p=8m$, the proposed correspondence is as follows.

Quantum $SO_3$ spin-Chern-Simons conjecture. The quantum $SO_3$ spin-Chern-Simons theory is, in fact, a TQFT and at the $1$ (mod $2$)-valued $SO_3$ level $2m-1$, it is isomorphic to the BM spin TQFT at level $p=8m$.

This conjecture proposes a gauge-theoretic realization of the knot-theoretic BM spin TQFTs. It is motivated by the analogous conjectural correspondence between quantum $SU_2$ Chern-Simons theory and the BHMV TQFT; the source presents it as an open conjecture because the relevant Feynman path integral is not yet mathematically well defined.