Gaussian likelihood-ratio fluctuation conjecture in the paramagnetic phase

Let $\Gamma$ be the set of parameter pairs $(\alpha,\beta)$ for which the replica-symmetric formula vanishes, and let $\Gamma'$ be its interior. For the likelihood ratio $L(\bm{Y};\beta)$, let $\mathbb{P}_0$ denote the null law and $\mathbb{P}_\beta$ the alternative law. Likelihood-ratio Gaussianity conjecture. For every $(\alpha,\beta)\inGamma'$, the log-likelihood ratio satisfies

$$\log L(\bm{Y};\beta)\rightsquigarrow {\mathcal{N}}\left(\pm \frac{1}{4}\log(1-\alpha\beta^2),-\frac{1}{2}\log(1-\alpha\beta^2)\right),$$

where the plus sign holds under the null law $\mathbb{P}_0$ and the minus sign under the alternative law $\mathbb{P}_\beta$. This predicts asymptotic normality throughout the interior of the undetectable region, extending the rigorously established region below the BBP threshold; the claim remains open for general priors.