Universality conjecture for critical one-dimensional random operators

Let $E$ be distributed according to the arcsine law, $U$ be uniform on $[0,1]$, and let $\mathcal{Z}$ be a standard two-sided Brownian motion started from $0$, independent of $E$ and $U$. Define

$$\rho(E)=\frac{1}{\sqrt{1-E^2/4}}\mathbf{1}_{|E|<2},\qquad \tau(E)=(\sigma\rho(E))^2,$$

and

$$S(t)=\exp\left(\frac{\mathcal{Z}_t}{\sqrt{2}}-\frac{|t|}{4}\right).$$

For each operator in the listed settings, consider the corresponding eigenvectors and their high-eigenvalue or small-parameter scaling regime. Universality conjecture. The eigenvectors have a scaling limit described by the limit in Theorem global_evector in the following settings: the critical continuous one-dimensional random Schrödinger operator, the Sine$_\beta$ operator as $\beta\to0$, the stochastic Airy operator as $\beta\to0$ for high eigenvalues, and the random Hill operator for high eigenvalues. This is a proposed extension of the discrete critical model's eigenvector limit to several one-dimensional random operators. The cited works indicate related settings, but the supplied text does not establish the conjectured common limit or state its resolution, so the status remains open.