Exact information-theoretic detection threshold for correlated SBM vs Erdos-Renyi pair

Fix constants $k\ge 2$, $\lambda>0$, and $\epsilon\in[-1/(k-1),1]$, and for each $n$ define

Assume $n$ is large enough that $p_{\mathrm{in}},p_{\mathrm{out}}\in[0,1]$. Let $[k]=\{1,\dots,k\}$.

Under $H_1$ (correlated SBM with latent matching), sample a parent graph $G^\star$ on latent vertices $[n]$ by first drawing labels $\sigma_1,\dots,\sigma_n\overset{i.i.d.}{\sim}\mathrm{Unif}([k])$, then conditionally on $\sigma$ sampling edges independently with

Generate $A^\star,B^\star$ by independent edge subsampling from $G^\star$ with parameter $s\in[0,1]$ (keep each parent edge independently in each child with probability $s$, and never add non-parent edges). Then draw a latent uniform permutation $\pi\in S_n$, independent of everything else, and observe

where $\pi(B^\star)$ is the relabeled graph. Let $P_n=P_{n,\lambda,k,\epsilon,s}$ be the law of observed $(A,B)$.

Under $H_0$, let $Q_n$ be the law where $A,B$ are independent Erdos-Renyi graphs $G(n,\lambda s/n)$ (equivalently, independent even after an unknown relabeling of one graph).