Critical-SNR conjecture for the 2-WSBM

In the 2-WSBM, let $N$ be the number of vertices, $k$ the community size, and let $\gamma$ denote the signal-to-noise ratio; assume $k=o(\log N)$. For $k=2$, the model is equivalent to the 1-P-REM with $M=\binom{N}{2}\approx N^2$ states, yielding the benchmark $\gamma_c=\sqrt{2}$. Critical-SNR conjecture. The critical signal-to-noise ratio for the 2-WSBM is

$$\gamma_c=\sqrt{\frac{2}{k-1}}.$$

The conjecture extends the exactly identified $k=2$ case to communities of size sublogarithmic in $N$ and predicts the recoverability threshold in this regime. Its proof is left for future work.