The low-degree polynomial hardness threshold conjecture for random k-SAT

Let $k$ be the clause width, let $\kappa$ be the clause-density factor, and let $\Phi_k(n,m)$ denote a random $k$-SAT formula with $n$ variables and $m$ clauses. Let Theorem~ and Theorem~ denote the hardness results stated above.

Low-degree polynomial threshold conjecture. Theorem~ (and Theorem~) holds for all $\kappa > 1$.

This conjecture would close the constant-factor gap between the paper's hardness bound and the best known algorithms, establishing the algorithmic phase transition at clause density $(1+o_k(1))2^k\log k/k$.