Decelle–Krzakala–Moore–Zdeborová Potts-model conjecture for k communities

Let $\mathcal{G}(n,a/n,b/n)$ be the planted partition model with $k$ communities, where $a>b$. Let $c(k)$ be a threshold depending only on $k$. Decelle–Krzakala–Moore–Zdeborová's Potts-model conjecture. For any $k$, there exists $c(k)$ such that

$$\frac{(a-b)^2}{a+(k-1)b}<c(k)$$

implies that the clustering problem cannot be solved;

$$c(k)<\frac{(a-b)^2}{a+(k-1)b}<k$$

implies that it is solvable but not in polynomial time; and

$$\frac{(a-b)^2}{a+(k-1)b}>k$$

implies that it can be solved in polynomial time. When $k\leq4$, $c(k)=k$, so the intermediate case does not occur; when $k\geq5$, $c(k)<k$. This conjecture predicts distinct information-theoretic and computational transitions for the multi-community model, motivated by phase transitions in the Potts model on trees. The source notes that non-reconstruction results are less precise for $k>2$ and gives no resolution of the conjecture.