Krzakala–Qiu–TAP conjecture for spherical Bayesian linear regression

Consider the Gaussian linear regression model with responses $\mathbf{y}\in\mathbb{R}^N$, design matrix $\mathbf{x}\in\mathbb{R}^{N\times p}$, coefficient vector $\boldsymbol{\beta}\in\mathbb{R}^p$, and noise level $\Delta>0$, with $\mathbf{y}=\mathbf{x}\boldsymbol{\beta}+\sqrt{\Delta}\mathbf{z}$ and posterior partition function $\mathcal{Z}_p$. Let $\alpha=N/p$, and let the prior be uniform on the sphere $\mathbb{S}^{p-1}(\sqrt{p})$. The free energy is $p^{-1}\log\mathcal{Z}_p$. Krzakala–Qiu–TAP conjecture. For all noise levels $\Delta>0$, the TAP free energy is asymptotically exact:

$$\lim_{p \to \infty} \frac{1}{p} \log \mathcal{Z}_p=\lim_{p \to \infty} \max_{\|a\| \leq \sqrt{p}}\left\{ -\frac{1}{2\Delta p}\|\mathbf{y}-\mathbf{x}a\|^2-\frac{\alpha}{2}\ln\left(1+\frac{1-\|a\|^2/p}{\Delta\alpha }\right)+\frac{1}{2}\ln\left(1-\frac{\|a\|^2}{p}\right)\right\}.$$

The conjecture extends the previously established high-temperature characterization to every positive noise level, including the low-temperature regime. Its resolution is relevant to the TAP approach and to the relationship between global TAP maximizers and posterior means; the supplied status evidence indicates that this extension remains open.