the proof of Lemma \[lemma:optSMV\], in order to rigorously prove Conjecture \[conj:2\], we need to show ([*i*]{}) the MMSE given by the replica analysis for the MMV case [@ZhuBaronKrzakala2017IEEE] is exact, ([*ii*]{}) the scalar channel noise variance $\Delta_v$ in Algorithm \[algo:AMP\_MMV\] corresponds to the MMSE given by replica analysis, and ([*iii*]{}) a result similar to Theorem 1 in Bayati and Montanari [@Bayati2011] holds. None of these three results exists in the prior art, so we do not foresee what exact conditions are needed for Conjecture \[conj:2\]. Proving these three results (and hence Conjecture \[conj:2\]) is beyond the scope of this work. Instead, we provide some intuition that explains why we believe Algorithm \[algo:metric\_opt\_MMV\] is optimal in the MMV scenario.

In the SMV case, $f(x_n|q_n)$ converges to the posterior $f(x_n|\y)$ in relaxed BP [@RanganGAMP2011ISIT]. As an extension to the MMV case, we intuitively think that $f(\x_n|\q_n)$ would converge to the posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$, based on two observations: ([*i*]{}) the measurement channels are i.i.d., so that it suffices to update the estimate of the channel by passing the messages $\mathcal{M}_{n\rightarrow m}(x_n^{(j)})$ and $\widehat{\mathcal{M}}_{m\rightarrow n}(x_n^{(j)})$ for different $j$ individually, and ([*ii*]{}) the super-symbols $\x_n$ are i.i.d., so that denoising each super-symbol individually accounts for 