The spherical-building conjecture for smooth-group actions on reductive groups

Suppose that $(\widetilde G_{k^{\mathrm a}},\Gamma_{k^{\mathrm a}})$ is quasisemisimple. Write $G^{\mathrm{red}}$ for the maximal pseudo-reductive quotient of $G$, which is reductive by the cited theorem. There are a unique subset $\mathscr{S}(G)^{\mathrm{red}}$ of $\mathscr{S}(G^{\mathrm{red}})$ and a map

$$i\colon\mathscr{S}(G)^{\mathrm{red}}\longrightarrow\mathscr{S}(\widetilde G)$$

with the following properties. For every split torus $S$ in $G^{\mathrm{red}}$, the subset $\mathscr{S}(S)\cap\mathscr{S}(G)^{\mathrm{red}}$ consists precisely of the rays through elements

$$\lambda\in {\mathbf X}_*(S)\otimes_{\mathbb Z}\mathbb R\setminus\{0\}$$

such that $P_G(\lambda)$ contains the unipotent radical of $G$, and the natural diagram relating $\mathscr{S}(S)$, $\mathscr{S}(\widetilde G)$), $\mathscr{S}(S)\cap\mathscr{S}(G)^{\mathrm{red}}$, and $\mathscr{S}(G)^{\mathrm{red}}$ commutes. Spherical-building conjecture. The map $i$ is a bijection from $\mathscr{S}(G)^{\mathrm{red}}$ onto

$$\mathscr{S}(\widetilde G)\cap\mathscr{S}(\widetilde G_{k^{\mathrm s}})^{\Gamma(k^{\mathrm s})}.$$

The existence of the set and map is described as accessible, with uniqueness then immediate; the claimed surjectivity is left for future work. Thus the bijectivity assertion is open in the source.