Kerov's positivity conjecture for character polynomials

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by

$$\Sigma^{\lambda}_{\pi}=K_{\pi}(R_2^{\lambda},R_3^{\lambda},\dots).$$

For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's conjecture. The coefficients of $K_k$, for $k\geq 1$, are non-negative integers.

This conjecture predicts a positivity property for the exact character formulas expressed through free cumulants. The source reports that it was formulated from numerical evidence; no resolution is supplied here.