Eventual sign periodicity conjecture for the weight two Euler characteristic of moduli spaces of curves

Let ${\mathcal M}_g$ denote the moduli space of smooth curves of genus $g$, and let $\chi_2({\mathcal M}_g)$ be its weight two compactly supported Euler characteristic. Eventual sign periodicity conjecture. For $g \geq 23$, the sign of $\chi_2({\mathcal M}_g)$ is $-1$ for $g \equiv 0,1 \pmod 4$ and $+1$ for $g \equiv 2,3 \pmod 4$. The displayed computations suggest an eventually periodic sign pattern, but no proof of this assertion is supplied.