Formal-map convergence conjecture for real analytic hypersurfaces in complex dimension two

Let $M$ and $M'$ be connected real analytic hypersurfaces through $0$ in $\mathbb C^2$, and suppose that $M$ is of finite type at some point. Let $F:(\mathbb C^2,0)\to(\mathbb C^2,0)$ be an invertible formal mapping sending $M$ into $M'$. Formal-map convergence conjecture. The formal mapping $F$ is convergent. The case in which $M$ is of finite type at $0$ is known; the conjecture concerns the case where finite type holds only away from the origin.