$\renewcommand{\!}{\mathbf} \renewcommand{\Ai}{\operatorname{Ai}}$ One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where $$\mathbf A(x, y)=\begin{cases} \frac{\Ai(x) \Ai'(y)-\Ai'(x) \Ai(y)}{x-y} & \text {if } x \neq y \\ \Ai'(x)^{2}-x \Ai(x)^{2} & \text {if } x=y \end{cases} \text{ for } \Ai(x) = \frac 1{\pi} \int_0^\infty \cos\Big(\tfrac 13 t^3 + xt\Big) \, d t$$ It's well known that the largest eigenvalue of an $n\times n$ GUE matrix (appropriately scaled) converges in distribution to $F_2$. Also, by the celebrated Baik-Deft-Johansson theorem, the length of the longest increasing subsequence of a random permutation $\in S_n$ (appropriately scaled) also converges in distribution to $F_2$. I'm sure there are many other examples of situations in which $F_2$ appears as the limiting distribution.

However, I have been trying to find references about how fast the things that converge to $F_2$ converge to $F_2$. My searches have turned up this paper: https://arxiv.org/pdf/0803.3408.pdf and this paper: https://arxiv.org/pdf/1901.05235.pdf, both of which seem to answer related questions but not exactly what I'm looking for.

Does anyone know the rates at which the largest eigenvalue of a GUE matrix and/or the longest increasing subsequence of a random permutation converge to $F_2$? If not exactly, are there references that have done numerical computations and put forward a conjecture about such rates?