The lifting conjecture for transfer systems on athbb{S}\mathrm{L}_2$\mathbb{F}_p$athbb{S}]

Let $G = \operatorname{SL}_2(\mathbb{F}_p)$ with $p > 5$, $p \neq 11$, and $p \equiv \pm 3 \mod 8$. Fix an arbitrary transfer system $\mathcal{R}$ on $G$, and let $(\mathcal{R}_D,\mathcal{R}_I,\mathcal{R}_U)$ be the triple obtained from $\mathcal{R}$ as described above. A lifting conjecture asserts that $(\mathcal{R}_D,\mathcal{R}_I,\mathcal{R}_U)$ is a split transfer system, and that $\mathcal{R}$ is lifted from this triple using the stated procedure.

This conjecture would provide a constructive method for exploring $N_\infty$ operads for groups that are not lossless, by reducing transfer systems to conjugacy data on the associated subgroup spaces. The source presents this as a speculative method; no resolution is supplied here.