The sparse block-model coupling conjecture below the square-root threshold

Let $0 \leq \delta \leq c$. Define $\mathbf{Q}=G(n,\kappa_1)$ for the constant kernel $\kappa_1\equiv c$, and let $\mathbf{P}=G(n,\kappa_2)$, where $\kappa_2$ equals $c+\delta$ on $[0,1/2]\times[1/2,1]$ and $c-\delta$ elsewhere. Two models are essentially equivalent if they admit a coupling $\mu$ such that

$$\mathbf{E}_\mu d_1(G,H)=o(n),$$

where $d_1$ is edit distance and $(G,H)\sim\mu$. The sparse block-model coupling conjecture. If $\delta<\sqrt{c}$, then $\mathbf{P}$ and $\mathbf{Q}$ are essentially equivalent. This is the below-threshold half of the conjecture attributed to the authors of [BR11]; the preceding text notes that the claim is known in the range $0\leq\delta\leq 1$ with $c\geq\delta$, while its full stated range is not resolved here.