Generalized Stolz conjecture for proper equivariant spin bordism

Let $\underline{E}\Gamma$ be the universal space for proper $\Gamma$-actions, and let $R^{\textup{spin}}_n(\underline{E}\Gamma)^\Gamma[\mathfrak B^{-1}]$ be the direct limit of

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

The associated index map is

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

Generalized Stolz conjecture. The index map $\Theta$ is an isomorphism. This is the proper-action analogue of the Stolz conjecture, replacing $B\Gamma$ by the universal proper $\Gamma$-space; its status is not specified in the source.