Goulden–Rattan conjecture on coefficients of Goulden–Rattan polynomials

Let $C_0=1$, $C_1=0$, and for $k\geq 2$ define $C_k$ on Young diagrams by

$$C_k^{\lambda}=\frac{24}{k(k+1)(k+2)}\lim_{s\to\infty}\frac{1}{s^k}\big(\Sigma_{k+1}(s\lambda)-R_{k+2}(s\lambda)\big).$$

Let $L_k$ be the polynomial with rational coefficients determined by $K_k-R_{k+1}=L_k(C_2,C_3,\ldots)$; these are the Goulden–Rattan polynomials. Goulden–Rattan's conjecture. The coefficients of the Goulden–Rattan polynomials are non-negative numbers with small denominators. The conjecture concerns a finer coefficient property than the already established positivity of Kerov character polynomials. The source gives no resolution of this general assertion; its later results address special linear and square coefficients, so the status is open.