Kerov's unified expansion conjecture for Kerov polynomial components

Let $K_r$ be the Kerov polynomial and write its component of weight $r-2k+1$ as $K_{r,r-2k+1}$. For a partition $u$, let $l( u)$ be its length, let $m_i( u)$ be the multiplicity of the part $i$, and set

$$\mathcal{R}_\mu=\prod_{i\ge 2} \frac{((i-1)R_i)^{m_i(\mu)}}{m_i(\mu)!}.$$

For a symmetric function $f_k$, write $f_k(\mu)$ for its value at the integral vector $\mu$. Kerov's unified expansion conjecture. For any $k\ge 1$ there exists an inhomogeneous symmetric function $f_k$, having maximal degree $4(k-1)$, such that

$$K_{r,r-2k+1}= \binom{r+1}{3} \sum_{|\mu|=r-2k+1} (l(\mu)+2k-2)!\, f_k(\mu)\,\mathcal{R}_\mu,$$

where this symmetric function is independent of $r$. This would unify the known formulas for the highest odd-weight components and, in particular, would give an $r$-independent description of lower components whose explicit forms are currently difficult to obtain.