Gaussian marginal conjecture for the pure-noise Gibbs measure

Let $\mathcal{S}^N$ be the sphere, let $H_0$ be the pure-noise Hamiltonian, and for $\beta>0$ let $\pi_{0,\beta}$ be the Gibbs measure with density proportional to $\exp(-\beta H_0(x))$ with respect to spherical volume. Write $\pi_{0,\beta}(x_1\in\cdot)$ for the marginal law of the first coordinate, let $\gamma$ be the law of a standard Gaussian, and let $\pi_{0,\beta}^+$ be $\pi_{0,\beta}$ conditioned on $\{x_1>0\}$. Gaussian marginal conjecture. For every $\beta>0$, almost surely with respect to the disorder, the marginal measures converge weakly:

$$\pi_{0,\beta}(x_1\in\cdot)\xrightarrow[N\to\infty]{}\gamma(\cdot).$$

In particular, $\pi_{0,\beta}^+$ satisfies Condition 2. This conjecture would imply an almost-sure recovery result for $k>2$ and $\alpha>\alpha_c(\infty)$, as well as the matching weak-recovery result for $k<2$, and is also of independent interest in statistical physics.