Shapiro–Tater scaling conjecture for discriminant roots and Painlevé II poles

Let $C_n(t,\lambda)$ be the secular polynomial of the quartic anharmonic oscillator, and let

$$D_n(t)=\operatorname{Disc}_{\lambda}(C_n)(t)$$

be its discriminant with respect to $\lambda$. Let $Y_n(t)$ denote the Vorob'ev–Yablonsky polynomials, whose zeros are the poles of the rational solutions of Painlevé II. Shapiro–Tater's scaling conjecture. The roots of the rescaled discriminant

$$D_n(s)=\operatorname{Disc}_{\lambda}(C_n)(n^{2/3}s)$$

and the roots of the rescaled Vorob'ev–Yablonsky polynomial $Y_n(n^{2/3}s)$ form two coinciding lattices as $n\to\infty$. This conjecture concerns the asymptotic similarity between oscillator eigenvalue degeneracies and the pole patterns of rational Painlevé II solutions; the supplied text presents it as a numerical observation and gives no resolution.