Hypergraph planted clique conjecture

Let $D$-uniform hypergraphs on the vertex set $[N]$ be hypergraphs whose hyperedges are incident on $D$ vertices. Under the null model, let $G\sim\mathcal{G}_D(N,p)$, where each hyperedge is included independently with probability $p$. Under the planted model, let $G\sim\mathcal{G}_D(N,p;K)$, where $K\geq D$ vertices are chosen uniformly at random, all $\binom{K}{D}$ hyperedges among them are included, and every remaining hyperedge is included independently with probability $p$. For a test $\psi_N:{\mathbb{H}}_{D,N}\mapsto\{0,1\}$, define

$$\mathcal{E}(\psi_N)=\frac{1}{2}\mathbb{E}_{H_0}[\psi_N(G)]+\frac{1}{2}\mathbb{E}_{H_1}[1-\psi_N(G)].$$

Hypergraph planted clique conjecture. Suppose that $p=1/2$ and that $D\geq 2$ is a fixed integer. If

$$\limsup_{N\to\infty}\frac{\log K}{\log\sqrt{N}}<1,$$

then for every sequence of tests $\{\psi_N\}_{N\geq 1}$ computable in time polynomial in $N^D$,

$$\liminf_{N\to\infty}\mathcal{E}(\psi_N)\geq\frac{1}{2}.$$

This is the assumed computational hardness of detecting a planted clique in a random $D$-uniform hypergraph below the $\sqrt{N}$ scale, and it serves as the primitive for the paper's lower bounds on polynomial-time adaptation. No resolution status is supplied in the source.