The Kikuchi spectral-threshold conjecture

Fix even $r$ and $\ell$ with $n\gg\ell\geq r/2$, and let $Z_S\sim\mathcal N(0,1)$ independently for each $r$-subset $S\subset[n]$. Define the Kikuchi matrix $M(\lambda)$, indexed by $\ell$-subsets of $[n]$, by

$$M(\lambda)_{I,J}=\begin{cases}\lambda+Z_{I\triangle J}&|I\triangle J|=r,\\\\0&\text{otherwise},\end{cases}$$

where $I\triangle J=(I\cup J)\setminus(I\cap J)$, and let $\lambda^\natural_{r,\ell}$ be the threshold at which eigenvalues pop out of the spectrum in the sense specified in the source. Kikuchi spectral-threshold conjecture. For fixed $r$,

$$n^{r/4}\lambda^\natural_{r,\ell}\to0$$

as $\ell\to\infty$, after taking $n\to\infty$. The conjecture concerns whether increasing the Kikuchi parameter reaches below the sum-of-squares scale; the threshold is characterized in part of the parameter range, but the stated asymptotic remains open.