Universal MAP estimator MSE conjecture

Let $x\in\mathbb{R}^N$ be the signal, $y=\Phi x+z$ the measurement vector, and $x^U_{MAP}$ the universal MAP estimator. Assume that $\Phi\in\mathbb{R}^{M\times N}$ is an i.i.d. Gaussian measurement matrix whose entries have mean zero and variance $1/M$, that Condition~ holds, that the aspect ratio $R>0$ in~, and that $z\in\mathbb{R}^M$ is i.i.d. zero-mean Gaussian noise with finite variance. Universal MAP estimator MSE conjecture. For every $\epsilon>0$, and for sufficiently large $N$,

$$\frac{E_{X,Z,\Phi}\left[\|x-x^U_{MAP}\|^2\right]}{N}<\frac{2E_{X,Z,\Phi}\left[\|x-E_X[x\mid y,\Phi]\|^2\right]}{N}+\epsilon.$$

The conjecture says that the universal MAP estimator achieves mean squared error at most twice the MMSE, up to an arbitrarily small additive term, in the large-system limit under the stated measurement and noise assumptions. Its status is not established by the supplied source context.