Carpentier's motif-counting conjecture for community recovery in the stochastic block model

Let $G=(V,E)$ be a connected graph with $(v_1,v_2)\notin E$. For any partition of $V$ into $\ell$ groups in which $v_1$ and $v_2$ belong to the same group, let $E^{\neq}$ be the set of edges of $E$ joining distinct groups, and set

$$r:=\frac{|E|}{|V|-2}.$$

Carpentier's motif-counting conjecture. If

$$|E^{\neq}|\geq r(\ell-1),$$

for every such partition, and if $q\asymp n^{-1/r}$ and $\lambda\geq_{\log}q^{1-\log_n(K)}$, then the communities in the stochastic block model can be recovered by an algorithm based on $P_G$, the count of occurrences of the motif $G$.

The conjecture identifies a sufficient combinatorial condition on a motif for counting its occurrences to achieve community recovery in the regime with more than $\sqrt n$ communities. The supplied text does not state whether this claim has been proved or disproved.