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Quadratic coefficients of Goulden-Rattan character polynomials

Goulden-Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities ($C_i$) which describe the macroscopic shape of the Young diagram. The Goulden-Rattan positivity conjecture states that the coefficients of these polynomials are positive rational numbers with small denominators. We prove a special case of this conjecture for the coefficient of the quadratic term $C_2^2$ by applying certain bijections involving maps (i.e., graphs drawn on surfaces).