Global rigidity lower bounds for the Pearcey process

Let $N(x)$ be the Pearcey process counting function, let $x_k$ denote its ordered points, and let $\rho$ be the parameter appearing in the deterministic asymptotics. For any $\epsilon>0$, the following two lower bounds hold:

Global rigidity lower-bound conjecture.

$$\lim_{s\to\infty}\mathbb P\left(\sup_{x>s}\left|\frac{N(x)-\left(\frac{3\sqrt{3}}{4\pi}x^{\frac{4}{3}}-\frac{\sqrt{3}\rho}{2\pi}x^{\frac{2}{3}}\right)}{\log x}\right|\geq\frac{4\sqrt{2}}{3\pi}-\epsilon\right)=1,$$ $$\lim_{k_0\to\infty}\mathbb P\left(\sup_{k\geq k_0}\frac{\left|\frac{3\sqrt{3}}{4\pi}x_k^{\frac{4}{3}}-\frac{\sqrt{3}\rho}{2\pi}x_k^{\frac{2}{3}}-k\right|}{\log k}\geq\frac{\sqrt{2}}{\pi}-\epsilon\right)=1.$$

These conjectured bounds assert sharp logarithmic-scale fluctuations for both the counting function and the locations of the Pearcey process points, complementing the corresponding upper bounds. Their status is not resolved in the supplied source context.