All-or-nothing conjecture for higher-order Gaussian fields

Consider a higher-order Gaussian generative field of order $bb$, with load $R$, channel capacity $\mathcal{C}(\sigma)$, overlap $m^\star$, and information rate $\mathcal{I}(\sigma)$. There exists an order $bb_0\geq 3$ such that, for $bb\geq bb_0$, higher-order Gaussian-field conjecture.

$$m^\star=\begin{cases}1 & R\leq \dfrac{\mathcal{C}(\sigma)}{\log 2},\\0 & R>\dfrac{\mathcal{C}(\sigma)}{\log 2},\end{cases}$$

and

$$\mathcal{I}(\sigma)=\begin{cases}R\log 2 & R\leq \dfrac{\mathcal{C}(\sigma)}{\log 2},\mathcal{C}(\sigma) & R>\dfrac{\mathcal{C}(\sigma)}{\log 2}. \end{cases}$$

This predicts an all-or-nothing transition at Shannon's channel-capacity threshold: perfect inference below the critical load and zero overlap above it. The source motivates the claim using heuristic comparisons and numerical validation, but does not provide a proof.