The noisy equitable stochastic block model detection conjecture

Let $\bT_\delta$ be the noise operator that rewires $\lfloor\delta n\rfloor$ swaps of a $d$-regular graph, and let $\sG_{n,d,k,\eta}^{\,\mathrm{eq}}$ be the equitable stochastic block model. Write $\widetilde{\sG}^{\,\mathrm{eq}}_{n,d,k,\eta,\delta}$ for the distribution of $\bT_\delta(\bG)$ when $\bG\sim\sG_{n,d,k,\eta}^{\,\mathrm{eq}}$. For fixed $\delta>0$, $\eta\in[-\frac{1}{k-1},1]$, $k\geq2$, and $d<d_{\mathrm{KS}}^{\,\mathrm{eq}}(\eta)$, with $k\mid(1-\eta)d$, the noisy equitable stochastic block model detection conjecture. There is no polynomial-time algorithm that, with high probability as $n\to\infty$, distinguishes $\widetilde{\sG}^{\,\mathrm{eq}}_{n,d,k,\eta,\delta}$ from $\sG_{n,d}$. The divisibility condition ensures that the equitable model is defined for an infinite sequence of values of $n$. This conjecture is used as a conditional hardness assumption for results on random regular graphs; the added rewiring noise is stated to be crucial to the conjecture's applicability.