Goulden–Rattan's numerical conjecture for the k=3 coefficients c_rho

Let $c_\rho^{(3)}$ be the coefficients in the monomial expansion of $f_3$. Goulden–Rattan's numerical conjecture. For $k=3$, the values of $2\cdot 6!\cdot 8!\,c_\rho^{(3)}$ are given by the table in the source. The table records computationally observed values supporting the preceding positivity conjecture for $f_3$; it is not itself a general formula for the coefficients.