Sigma-Laplacian recovery conjecture for the Gaussian planted submatrix model

Consider the Gaussian Planted Submatrix model in the setting of Theorem~, with signal parameter $\beta$, signal vector $\bm{x}$, and leading eigenvector $\bm{v}_{1}(\bm{L}_{\sigma}(\widehat{\bm{Y}}))$ of the corresponding $\sigma$-Laplacian. Gaussian planted submatrix recovery conjecture. If $\beta>0.76$, then

$$|\langle \bm{v}_{1}(\bm{L}_{\sigma}(\widehat{\bm{Y}})),\bm{x}\rangle|$$

converges in probability to a strictly positive deterministic number. This conjecture would establish weak recovery beyond the threshold achieved by the direct spectral algorithm; the paper gives numerical evidence and leaves the assertion open.