Maillard et al.'s replica prediction for Bayes risk

Consider the generalized linear model with orthogonally invariant design, and let $\widehat{\beta}_{\mathrm{B}}=\mathbb{E}[\beta_*\mid X,y]$ be the posterior mean estimator. Let $(\widehat q_x^*,\widehat q_z^*,\gamma_x^*,\gamma_z^*,q_x^*,q_z^*)$ be a tuple in

$$[0,\infty)^4\times[0,\rho]\times[0,Q_z]$$

solving the replica saddle point equations, where $\rho$ is the signal variance and $Q_z$ is defined by the model. Replica Bayes-risk conjecture. There exists such a tuple for which

$$\lim_{d\to\infty}\frac{1}{d}\mathbb{E}\left[\|\beta_* - \widehat\beta_{\mathrm{B}}\|^2\right]=\rho-q_x^*.$$

This conjecture predicts the asymptotic Bayes mean-squared error through the replica saddle point equations. The source later relates their solutions to Bayes-GVAMP state-evolution fixed points, but does not establish the asserted asymptotic risk formula in general.