Near-optimal Hilbert-cube points are close to binary points

Let $A\in(\mathbb{R}^N)^{\otimes p}$ have independent entries distributed as $\mathcal{N}(0,N^{-(p-1)})$, and let $\mathcal{B}_N\subset\mathcal{H}_N$ denote the binary and Hilbert-cube domains used for the p-spin model. Hilbert-cube localization conjecture. For every $\epsilon>0$, there exists $\delta>0$ such that, with probability at least $1-\exp(-cN)$ for some $c>0$ and all sufficiently large $N$, every $u\in\mathcal{H}_N$ satisfying

$$A(u)/N\leq(1-\delta)\mathbb{E}[\eta_N]$$

also satisfies

$$\min_{v\in\mathcal{B}_N}\|u-v\|_2\leq\epsilon\sqrt{N}.$$

If true, this would reduce the conjectured OGP for near-optimal points in the Hilbert cube to the established binary-domain OGP; the paper explicitly presents it as an interesting open problem.