Kerov's Q-positivity conjecture for Kerov polynomial components

Define $Q_0=1$, $Q_1=0$, and for $n\ge 2$ let

$$Q_n=\sum_{|\mu|=n}(l(\mu)-1)!\,\mathcal{R}_\mu,$$

where $l(\mu)$ is the length of $\mu$ and $\mathcal{R}_\mu=\prod_{i\ge2}((i-1)R_i)^{m_i(\mu)}/m_i(\mu)!$. Kerov's Q-positivity conjecture. For $k\ge 1$ the coefficients of $K_{r,r-2k+1}$ in terms of the $Q_i$'s are nonnegative rational numbers. The source explains that C-positivity implies Q-positivity, which implies R-positivity; Q-positivity is proved for $k=1$ and remains conjectural in general.