Stanley's coefficient conjecture for Kerov character polynomials

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of $R_2^i$. Stanley's coefficient conjecture. For every $i\geq 1$,

$$[R_2^i]\Sigma_{2i+3,4}=\frac{1}{540}i(i+1)^3(i+2)^3(i+3)(2i+3).$$

This conjecture was communicated by Biane as a private conjecture of Stanley concerning higher-order terms. It is proved later in the paper from the stated corollary, so the conjecture is solved.