The hard-but-detectable conjecture for sparse PCA

Let $X$ have prior distribution $P_0$, and consider the rank-one symmetric matrix estimation model

$$Y_{i,j}=\sqrt{\frac{\lambda}{n}}X_iX_j+Z_{i,j},$$

where $Z_{i,j}$ are standard Gaussian noises. Assume $\mathbb{E}_{P_0}[X]=0$ and $\mathbb{E}_{P_0}[X^2]=1$. Let $\lambda_c$ be the critical value defined by the model's fixed-point criterion, and let $\operatorname{MSE}$ denote mean square error and $\operatorname{DMSE}$ the error of the dummy estimator. The hard-but-detectable conjecture. If $\lambda_c<1$, then achieving an $\operatorname{MSE}$ better than the dummy estimator is computationally hard for $\lambda\in(\lambda_c,1)$. This conjecture extends the hard-but-detectable phenomenon from sparse PCA and related community-detection models; the stated proposition establishes information-theoretic achievability in this interval, but does not establish efficient achievability or computational hardness.