Invertibility transfer from bosonic convolution to fermionic tensor product

Let $\tilde\beta$ be the bosonic-shadow theory and $\tilde\alpha$ the corresponding fermionic anomaly theory, with a conjectural defect $\tilde z_c$ relating them. Invertibility-transfer conjecture. If $\tilde\beta$ is invertible under the convolution product, then the existence of $\tilde z_c$ transfers this invertibility to $\tilde\alpha$, which is invertible under the regular tensor product. This conjecture is a step toward the broader bosonization picture and would establish invertibility of the fermionic anomaly theory from the bosonic-shadow construction. The source does not state a resolution.