Kerov's numerical conjecture for the k=3 coefficients a_rho

Let $a_\rho^{(3)}$ be the coefficients in the monomial expansion of $g_3$. Kerov's numerical conjecture. For $k=3$, the values of $500\cdot5!\cdot7!\,a_\rho^{(3)}$ are given by the table in the source. These finite computational values support the preceding conjecture that the coefficients of $g_k$ are positive rational numbers.