Powered spectral methods achieve weak recovery in the stochastic block model

Let $\mathrm{SBM}(n,p,Q/n)$ be a stochastic block model in which vertices from every community have the same expected degree. Let $A^{(r)}$ and $B^{(r)}$ be the powered adjacency and powered nonbacktracking matrices, respectively, and let $d$ be the graph diameter. Weak recovery means recovering the communities with nontrivial accuracy.

Powered spectral recovery conjecture. Choose $p$ and $Q$ such that vertices from every community have the same expected degree and there is an efficient algorithm that solves weak recovery on $\mathrm{SBM}(n,p,Q/n)$. There exists $0<r_0<1$ such that, for every constant $r<r_0$, the algorithm that finds the second-largest-eigenvalue eigenvector of $A^{(rd)}$ or $B^{(rd)}$ and partitions vertices by the median of the sums of entries corresponding to incident edges solves weak recovery on $\mathrm{SBM}(n,p,Q/n)$.

The conjecture extends the proposed diameter-scale powering approach to the stochastic block model at smaller constant fractions of the diameter. The source gives no resolution.