Stolz conjecture for the localized spin bordism group

Let $(L,f,h)$ represent an element of $R^{\textup{spin}}_n(B\Gamma)$, where $L$ is an $n$-dimensional spin manifold with boundary carrying a positive-scalar-curvature metric $h$, together with a map $f:L\to B\Gamma$. Let $\mathfrak B$ be the Bott manifold used in the source, and let $R^{\textup{spin}}_n(B\Gamma)[\mathfrak B^{-1}]$ be the direct limit of

$$R^{\textup{spin}}_n(B\Gamma)\xrightarrow{\times\mathfrak B}R^{\textup{spin}}_{n+8}(B\Gamma)\xrightarrow{\times\mathfrak B}R^{\textup{spin}}_{n+16}(B\Gamma)\to\cdots.$$

The associated index map is

$$\theta:R^{\textup{spin}}_n(B\Gamma)[\mathfrak B^{-1}]\to KO_n(C^*_r(\Gamma;\mathbb R)),$$

with $(L,f,h)$ sent to $\operatorname{Ind}(D_{L_\Gamma})$. Stolz conjecture. The index map $\theta$ is an isomorphism. This conjecture asserts that the localized spin bordism data with positive scalar curvature is completely detected by the real higher index; its status is not specified in the source.