The conjectured optimal efficient mean square error for symmetric low-rank estimation

Let $X$ have prior distribution $P_0$, and consider the symmetric rank-one matrix estimation model with signal-to-noise parameter $\lambda$. For $\eta>0$, define

$$q^0_\eta=\eta,\qquad q^{t+1}_\eta=\mathbb{E}_{P_0}[X^2]-\operatorname{mmse}(\lambda q^t_\eta),$$

and set

$$\widetilde q=\lim_{\eta\to0}\lim_{t\to\infty}q^t_\eta.$$

Here $\operatorname{mmse}$ denotes the scalar minimum mean square error. The efficient-estimation conjecture. For the model, the best mean square error achievable by an efficient algorithm is

$$\mathbb{E}_{P_0}[X^2]^2-\widetilde q^2.$$

This conjecture proposes that the state-evolution limit determines the optimal performance of efficient algorithms, addressing both whether the dummy-estimator error can be beaten efficiently and whether the information-theoretic minimum error can be attained; the source does not give a resolution.