Powered adjacency spectral clustering achieves weak recovery above the SBM threshold

Let $\mathrm{SBM}(n,a,b)$ be the two-community stochastic block model, let $A^{(r)}$ be its powered adjacency matrix, and let $w$ be an eigenvector associated with the eigenvalue of second-largest absolute value. Weak recovery means recovering the communities with nontrivial accuracy.

Powered adjacency recovery conjecture. Let $a,b\geq 0$ satisfy $(a-b)^2>2(a+b)$, let $r=\Theta(\sqrt{\log n})$, and let $G\sim\mathrm{SBM}(n,a,b)$. If $w$ is the eigenvector of $A^{(r)}$ with eigenvalue of second-largest absolute value, then dividing the vertices into those with entries in $w$ above the median and those with entries below the median solves weak recovery.

The inequality is the stated spectral detectability condition, and the conjecture claims that a square-root-logarithmic power suffices. The source gives no resolution.