Powered adjacency eigenvectors achieve weak recovery in the geometric block model

Let $\mathrm{GBM}(n,s,t)$ be the geometric block model, let $G$ be a graph from this model, let $d=\operatorname{diam}(G)$, and let $A^{(r)}$ denote the powered adjacency matrix. For sufficiently small $\varepsilon>0$, set $r=\varepsilon d$. Weak recovery is the task of recovering the communities with nontrivial accuracy.

Powered adjacency conjecture. Let $s,t\geq 0$ and suppose $\mathrm{GBM}(n,s,t)$ has a giant component. Taking the second-largest eigenvector of $A^{(r)}$, with $r=\varepsilon\cdot\operatorname{diam}(G)$ for sufficiently small $\varepsilon$, and rounding it by signs or by the median solves weak recovery in $\mathrm{GBM}(n,s,t/\sqrt{n})$ whenever weak recovery is solvable.

The conjecture asserts that graph powering followed by a spectral partition reaches the efficient weak-recovery threshold in the geometric block model. No resolution is supplied in the source.