Bordism invariance conjecture for the instanton line bundles

Let $Y$ be the manifold and let $\rho$ and $\sigma$ denote the representations used to define the instanton line bundles. In the above construction, a bordism $\hat{W}$ connects the manifolds $\hat{\Sigma}_l$ and $\hat{\Sigma}_l'$, and $\mathcal{L}_{\hat{\Sigma}_l}(\rho,\sigma)$ and $\mathcal{L}_{\hat{\Sigma}_l'}(\rho,\sigma)$ are the associated line bundles.

Bordism invariance conjecture. The bordism $\hat{W}$ between $\hat{\Sigma}_l$ and $\hat{\Sigma}_l'$ gives a natural isomorphism

$$\mathcal{L}_{\hat{\Sigma}_l}(\rho,\sigma)\cong \mathcal{L}_{\hat{\Sigma}_l'}(\rho,\sigma)$$

which is compatible with the gluing maps.

The conjecture is intended to provide invariance of the instanton line-bundle construction under the bordism described above. The paper does not prove this compatibility, so its status remains open.