Log-free performance conjecture for the even-order Kikuchi algorithms

Fix an integer $p$ and let $\ell$ be constant independently of $n$. Log-free performance conjecture. There exists a constant $c_p(\ell)>0$, with $c_p(\ell)\to 0$ as $\ell\to\infty$ for fixed $p$, such that whenever $\lambda\ge c_p(\ell)n^{-p/4}$, the detection and recovery algorithms at level $\ell$, which run in time $n^{O(\ell)}$, achieve strong detection and strong recovery, respectively. Removing the logarithmic factor would match the conjectured power of these polynomial-time algorithms with the guarantees suggested by the preceding results.