The planted dense subgraph recovery conjecture

Let $G_n\sim\mathsf{PDS}(n,k_n,p_n,q_n)$, where the planted support has size $k_n$. Exact recovery means outputting the planted support. If $k_n=\omega(\sqrt n)$ and

$$\limsup_{n\to\infty}\log_n\frac{k_n^2(p_n-q_n)^2}{nq_n(1-q_n)}<0,$$

the planted dense subgraph recovery conjecture. No polynomial-time algorithm $\mathcal{A}:G\to{[n]\choose k_n}$ can achieve exact recovery asymptotically.

The conjecture concerns the computational threshold for recovering the planted dense subgraph, which is asserted to be substantially stronger than the threshold for detection. The surrounding discussion attributes this conjectured threshold to prior work, but the supplied text gives no resolution.