The fixed-point conjecture for the partition complex at elementary abelian subgroups

Let $\Cwiggle=C_{U\kern-.5pt\left(p^{k}\right)}\left(\Delta_{k}\right)/\left(\Delta_{k}\times S^{1}\right)$, where $C_{U\left(p^k\right)}(\Delta_k)$ is the centralizer of $\Delta_k$ in $U(p^k)$. Let $T\operatorname{GL}_{k}\left(\mathbb{F}_{p}\right)^{\diamond}$ denote the unreduced suspension of the Tits building for $\operatorname{GL}_k(\mathbb{F}_p)$. Fixed-point conjecture. There is a homotopy equivalence

$$\left(\Lcal_{p^{k}}\right)^{\Delta_{k}} \simeq \Cwiggle_{+} \wedge T\operatorname{GL}_{k}\left(\mathbb{F}_{p}\right)^{\diamond}.$$

This conjecture is a consequence that the authors derive from the branching conjecture, and it is compared with the fixed-point information obtained from theorems on $\Gamma_k$- and $\Delta_k$-fixed points. Its status remains open in the supplied text.