The Secret Leakage Planted Clique Conjecture

Let $\rho$ be a distribution on $k$-subsets of $[n]$, and let $p_\rho(s)=\mathbb P_{S,S'\sim\rho^{\otimes 2}}[|S\cap S'|=s]$ be the intersection-size probability mass function. Let $\textsc{pc}_\rho(n,k,1/2)$ be planted clique with the planted set sampled from $\rho$. Secret Leakage Planted Clique Conjecture. If $\rho$ is one of the specified distributions, and there are $p_0=o_n(1)$ and a constant $\delta>0$ such that, for every parameter $d=O_n((\log n)^{1+\delta})$,

$$p_\rho(s)\le p_0\cdot\begin{cases}2^{-s^2}&\text{if }1\le s^2<d,\\s^{-2d-4}&\text{if }s^2\ge d,\end{cases}$$

then no polynomial-time algorithm solves $\textsc{pc}_\rho(n,k,1/2)$. This conjecture proposes that suitably small intersection tails do not make secret-leakage planted clique efficiently solvable.