The Bernoulli-number formula for the orbifold Euler characteristic of moduli space

Let $g\geq 2$ be an integer. Let $e_g(z_0)$ be the genus-$g$ generating function, written in partial fractions with coefficients $q_\ell^{(0)}$, let $C^{(g)}$ be the associated constant, let $B_{2g}$ denote the even Bernoulli number, and let $M_g$ be the moduli space of genus-$g$ Riemann surfaces. Then

Bernoulli-number conjecture. The coefficients satisfy

$$-C^{(g)}=\sum_{\ell=0}^{3g-3}q_\ell^{(0)}=\frac{B_{2g}}{4g(g-1)}=\chi_{\mathrm{orb}}(M_g).$$

This conjecture identifies the constant term from the map-counting generating function with the orbifold Euler characteristic of the moduli space of genus-$g$ Riemann surfaces. The source reports numerical verification for $2\leq g\leq100$ and attributes the conjecture to earlier work, but gives no proof or resolution.