Gal's arithmetic positivity conjecture for Kerov polynomials

Let $p$ be an odd prime, and let $\Sigma_{p+1}$ be the Kerov character polynomial on a cycle of length $p+1$, with $R_i$ denoting the free cumulants. Gal's conjecture. The expression

$$\frac{\Sigma_{p+1}-R_{p+2}+R_3}{p}$$

is a polynomial in $R_2,R_3,\dots$ with non-negative integer coefficients.

The conjecture was formulated by Światosław Gal from numerical calculations and extends the arithmetic positivity results proved in the preceding proposition for related prime-indexed expressions. The source gives no resolution.