The k-planted clique detection conjecture

Fix a constant $p\in(0,1)$. Let $E$ be a fixed known partition of $[n]$ into $k_n$ equally sized subsets, and under the alternative choose exactly one planted clique vertex uniformly from each part. Let $k\textsc{-PC}_D(n,k_n,p)$ denote the resulting detection problem, with hypotheses $H_0$ and $H_1$. For randomized polynomial-time algorithms $A_n:G_n\to\{0,1\}$ and positive-integer sequences $k_n$ satisfying

$$\limsup_{n\to\infty}\log_n k_n<\frac{1}{2},$$

the $k$-planted clique conjecture. If $G$ is an instance of $k\textsc{-PC}_D(n,k_n,p)$, then

$$\liminf_{n\to\infty}\left(\mathbb{P}_{H_0}[A_n(G)=1]+\mathbb{P}_{H_1}[A_n(G)=0]\right)\geq 1.$$

This is a planted-clique hardness assumption with secret leakage: the algorithm knows the partition, while the planted set contains one uniformly selected vertex from each part. The supplied text gives no evidence that the conjecture has been resolved.