Comparison conjecture for the extremal locations $X$ and $X_{\min}$

Let $x_1<x_2<0$, and let $X(x)$ and $X_{\min}(x)$ denote the corresponding extremal location functions for real solutions of the first Painlevé equation. Comparison conjecture.

$$0<X(x_2)-X(x_1)<X_{\min}(x_2)-X_{\min}(x_1).$$

The source presents this as an open question motivated by numerical behaviour and explicitly asks whether it can be proved or disproved.