Index-difference conjecture for psc-metrics on spin manifolds with fb-singularities

Let $X_{\Sigma}$ be a spin manifold with a $(\eta\text{-}\mathrm{fb})$-singularity of dimension $n\geq 9$. Assume that $\beta X\neq\varnothing$ and that ${\mathcal R}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$, with base point $g_0\in{\mathcal R}^{\mathrm{psc}}(X_{\Sigma})$. Let $KO^{\eta\text{-}\mathrm{fb}}_{q+n+1}=KO_{q+n+1}(\mathbb{CP}^{\infty})$ denote the relevant $KO$-theory group, and let $\mathbf{KO}^{\eta\text{-}\mathrm{fb}}$ be its representing spectrum. Index-difference conjecture. There is an index-difference map

$$\mathsf{inddiff}_{g_0}^{\eta\text{-}\mathrm{fb}}\colon {\mathcal R}^{\mathrm{psc}}(X_{\Sigma})\to\Omega^{\infty+n+1}\mathbf{KO}^{\eta\text{-}\mathrm{fb}},$$

which induces a non-trivial homomorphism

$$\big(\mathsf{inddiff}_{g_0}^{\eta}\big)_*\colon\pi_q({\mathcal R}^{\mathrm{psc}}(X_{\Sigma}))\to KO^{\eta\text{-}\mathrm{fb}}_{q+n+1}$$

when the target group $KO^{\eta\text{-}\mathrm{fb}}_{q+n+1}=KO_{q+n+1}(\mathbb{CP}^{\infty})$ is non-trivial. This conjecture predicts non-trivial homotopy detected by index-difference invariants in spaces of positive-scalar-curvature metrics on manifolds with fibered-boundary singularities; its status is not resolved in the supplied source.