Approximate symmetry conjecture for real Painlevé I solutions

Let $y_0\in\mathbb R$ and $y_l\leq y_0$ define a first solution of the first Painlevé equation, and let the second solution be defined by the data $-y_l$ and $-y_0$. Let $\Delta$ be the difference between their initial slopes. Approximate symmetry conjecture. The integer part of this difference is zero:

$$[\Delta]=0.$$

The claim is based on numerical values of the slope-difference function and expresses an approximate symmetry under simultaneous sign reversal of the initial and minimum values; it remains unproved in the source.