Coleman–Oort conjecture, arithmetic version

Let $\mathcal{M}_g$ denote the moduli space of smooth curves of genus $g$. A curve in $\mathcal{M}_g$ has a CM Jacobian when its Jacobian is a CM abelian variety. Coleman–Oort conjecture, arithmetic version. For sufficiently large genus $g$, specifically $g\geq 8$, $\mathcal{M}_g$ contains only finitely many curves with a CM Jacobian. This is the arithmetic formulation of the predicted rarity of CM Jacobians in the moduli space of curves; the supplied text gives no resolution status.