Powered adjacency spectral clustering is optimally accurate in the geometric block model

Let $A^{(r)}$ denote the powered adjacency matrix, let $d$ be the graph diameter, and let $m,s,t>0$ with $0<r'<1/2$. Weak recovery means recovering the communities with nontrivial accuracy.

Powered adjacency conjecture. For all $m>0$, $s>0$, $t>0$, and $0<r'<1/2$, the algorithm that finds the eigenvector of $A^{(r'd)}$ with the second-largest eigenvalue and then divides the vertices according to whether the entries corresponding to incident edges lie above or below the median recovers the communities on $\mathrm{GBM}(n,s,t)$ with optimal accuracy.

This is presented as the adjacency-matrix analogue of the powered nonbacktracking claim, motivated by the abundance of short cycles in the geometric block model. No resolution is stated.