Eigenfunction and noise-profile localization conjecture

Let $d\in\{1,2,3\}$. For each $n\ge 1$, let $\varphi_{n,L}$ be the corresponding eigenfunction, let $U_{n,L}\in[-L/2,L/2]^d$ be a point where $|\varphi_{n,L}|$ reaches its global maximum, and let $Q$ be the unique radial positive solution on $\mathbb R^d$ of

$$-\Delta Q-Q^3=-Q.$$

Set $a_L:= (C_d\log L)^{\frac1{2-\frac d2}}$ and $\psi_*(x):=Q(x)/\|Q\|_{L^2}$. Eigenfunction localization conjecture. For any $n\ge1$, the following convergence holds in probability as $L\to\infty$:

$$\left(\frac1{a_L^{d/4}} |\varphi_{n,L}|\left(U_{n,L}+\frac{x}{\sqrt{a_L}}\right),x\in\mathbb R^d\right)\Rightarrow\psi_*;$$ $$\left(\frac1{a_L}\xi\left(U_{n,L}+\frac{x}{\sqrt{a_L}}\right),x\in\mathbb R^d\right)\Rightarrow-\frac{\psi_*^2}{\|\psi_*\|_{L^4(\mathbb R^d)}^2}\sqrt{\frac{2d}{C_d}}.$$

This predicts that, after centering at a maximum and rescaling by the localization scale $a_L^{-1/2}$, every fixed eigenfunction has the universal profile $\psi_*$, while the rescaled noise converges to the corresponding squared profile. The source gives the optimizer characterization of $Q$ and the identity $\|Q\|_{L^4}^4=2d/C_d$, but does not state that this conjecture has been proved or refuted.