1 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montreal, QC H3G 1M8 Canada.
2 Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. centre ville, Montreal, QC H3C 3J7 Canada.
3 SISSA, International School for Advanced Studies, via Bonomea 265, Trieste, Italy.
4 INFN sezione di Trieste, Trieste, Italy.
5 School of Mathematics, University of Bristol, Fry Building, Bristol, BS8 1UG UK.

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of $t\in C$ for which the spectrum of the quartic anharmonic oscillator in the complex plane $\begin{array}{c}\frac{{\text{d}}^{2}y}{\text{d}{x}^2}-\left(x^4,+,t,x^2,+,2,J,x\right)y=\Lambda y,\end{array}$ with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.