Bosonization conjecture for anomaly theories

Let $G$ be a fermionic symmetry acting on a super modular tensor category, with anomaly $\tilde\alpha$. Let $F_{\tilde\alpha}$ be the Dijkgraaf–Witten-type theory obtained by summing $\tilde\alpha$ over twisted spin structures, and let $F_{\tilde\beta}$ be the analogous theory for the bosonic shadow of the anomaly. Let $z_c$ be the bosonization defect. Bosonization conjecture. As theories of twisted spin manifolds, $z_c$ extends from an $(F_{\mathrm{Spin}},F_B)$-defect to an $(F_{\tilde\alpha},F_{\tilde\beta})$-defect. The conjecture is intended to explain why the bosonic shadow construction captures the anomaly of the fermionic theory and would imply that the associated bosonic-shadow partition function is a twisted spin cobordism invariant. The source does not state a resolution.