C-positivity conjecture for the graded Kerov character polynomials

Let $R(x)=1+\sum_{i\geq 2}R_ix^i$, and define $C_m$ by $C_0=1$, $C_1=0$, and

$$C_m=\sum_{\substack{j_2,j_3,\ldots\geq 0\\2j_2+3j_3+\cdots=m}}(j_2+j_3+\cdots)!\prod_{i\geq 2}\frac{((i-1)R_i)^{j_i}}{j_i!},\qquad m\geq 2.$$

For $n\geq 0$, let $\Sigma_{k,2n}$ be the sum of the terms of weight $k+1-2n$ in $\Sigma_k$. A polynomial in the $C_m$ is C-positive when all its nonzero coefficients are positive. C-positivity conjecture. For $n\geq 1$ and $k\geq 2n-1$, $\Sigma_{k,2n}$ is C-positive. The conjecture is based on computations through $k=25$; the explicit formula for $\Sigma_{k,2}$ gives the first nontrivial case, while positivity for general $n$ and $k$ remains open.