Odd-order Kikuchi Hessian optimality conjecture

Let $p$ be odd and fix an integer $\ell\in[\lfloor p/2\rfloor,n-\lceil p/2\rceil]$. For $S\subseteq[n]$ with $|S|=\ell$ and $T\subseteq[n]$ with $|T|=\ell+1$, define the rectangular symmetric-difference matrix $M\in\mathbb{R}^{\binom{n}{\ell}\times\binom{n}{\ell+1}}$ by

$$M_{S,T}=\begin{cases}Y_{S\triangle T},&|S\triangle T|=p,\\0,&\text{otherwise.}\end{cases}$$

Let $u$ be a unit-norm top left singular vector of $M$, let $v=M^\top u$ be the corresponding top right singular vector, and define the recovery output $\widehat{x}\in\mathbb{R}^n$ by $\widehat{x}_i=\sum_{S,T}u_Sv_T\mathbf{1}_{S\triangle T=\{i\}}$. Odd-order Kikuchi Hessian optimality conjecture. In the Rademacher-spiked tensor model with odd $p\ge3$, if

$$\lambda\gg \ell^{-(p-2)/4}n^{-p/4},$$

then there is a threshold $\tau=\tau(n,p,\ell,\lambda)$ such that thresholding the top singular value of $M$ at $\tau$ achieves strong detection, and the stated singular-vector algorithm achieves strong recovery. The conjecture asserts that this odd-order Kikuchi Hessian algorithm matches the performance of sum-of-squares; only sub-optimal results for this algorithm are known in the paper, while optimal results are proved for a related algorithm.