Following this question: Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?.

We know that for $\lambda_N\le \lambda_{N_1}\le \dots le\lambda_1$ (eigenvalues of GOE matrix)

$$ \lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dotsc, N^{2/3}(\lambda_{N-k+1}-2)\le s_k)=F_{\beta, k}(s_1,\dotsc, s_k). $$

Can we say that for any $\epsilon>0$, there exists a constant $c>0$ such that $$ P(N^{2/3}(\lambda_N-\lambda_{N-1})\ge c)\ge 1-\epsilon? $$