Pointwise ordering conjecture for the maximal and minimal Painlevé I solutions

Let $y_{\max}(x_0;x)$ and $y_{\min}(x_0;x)$ be the two real solutions with pole at $x_0\leq0$, with $x$ in the interval of existence of $y_{\min}(x_0;x)$. Ordering conjecture.

$$y_{\max}(x_0;x)<y_{\min}(x_0;x).$$

The inequality was numerically checked for $x_0\in[-3,0]$ and is conjectured for every $x_0\leq0$; no proof is given.