AMP-UD recovery conjecture for stationary ergodic signals

Consider a linear inverse problem in which an input signal ${\bf x}$, generated by a stationary ergodic source, is estimated from noisy measurements ${\bf y}$ and measurement matrix ${\bf A}$. AMP applies the proposed AMP-UD denoiser, based on context quantization and Gaussian-mixture-model-based i.i.d. subsequence denoising, within its iterations. AMP-UD recovery conjecture. Under some technical conditions, AMP-UD achieves the minimum mean-square error (MMSE). This is a more detailed formulation of the paper's AMP-UD conjecture; the result is presented without a proof, and the relevant technical conditions are not specified.