Tracy–Widom limit conjecture for the centered statistic $T_1(Y)$

Let $T_1(Y)$ be the statistic in the separability test, and let $\gamma_0$ and $E_+$ be the quantities defined in the paper's equation (sec4.3.eq3). Let $TW_1$ denote the Tracy–Widom distribution of type 1. Tracy–Widom limit conjecture. Under the same conditions as in Theorem sec4.3.thm1,

$$\gamma_0n^{2/3}(T_1(Y)-E_+)\Rightarrow TW_1.$$

The conjecture proposes replacing the random centering $\hat E_+$ by the deterministic quantity $E_+$ in the asymptotic null distribution. It is motivated by convergence of $\hat E_+$ to $E_+$ in probability and empirical evidence, but remains unproved in the cited context.