The stronger secret-leakage planted clique conjecture

Let $\rho$ be sufficiently symmetric, and let $p_\rho(s)$ denote the intersection-size distribution for two independent sets sampled from $\rho$. Stronger Secret Leakage Planted Clique Conjecture. There is no polynomial-time algorithm solving $\textsc{pc}_\rho(n,k,1/2)$ if some function $w(n)=\omega_n(1)$ exists such that the tail bounds of the Secret Leakage Planted Clique Conjecture hold for every $d\le w(n)$. This strengthens the usual conjecture by requiring the intersection-tail bounds only up to an arbitrary diverging degree parameter rather than up to polylogarithmic degree.