Asymptotic independence of extreme eigenvalues in Gaussian beta-ensembles

For $\beta\in\{1,2,4\}$, let $X_{\beta,n}^{\min}$ and $X_{\beta,n}^{\max}$ be the centred and rescaled smallest and largest eigenvalues of a random matrix from the Gaussian $\beta$-ensemble, defined by

$$X_{\beta,n}^{\min}:=\sqrt{2}n^{1/6}(\min X_i+\sqrt{2n}),\qquad X_{\beta,n}^{\max}:=\sqrt{2}n^{1/6}(\max X_i-\sqrt{2n}).$$

Asymptotic independence conjecture. For every $x,y\in\mathbb{R}$,

$$\lim_{n\to\infty}\Big(\mathbb{P}[X_{\beta,n}^{\min}\le x, X_{\beta,n}^{\max}\le y]-\mathbb{P}[X_{\beta,n}^{\min}\le x]\mathbb{P}[X_{\beta,n}^{\max}\le y]\Big)=0.$$

This conjecture would provide the joint limit theorem for the smallest and largest eigenvalues needed to analyze the maximum absolute eigenvalue and the critical intersection problem for classical and Schatten $p$-balls. The source notes that the assertion is known for $\beta=2$, while for $\beta\in\{1,4\}$ it remains open.