Centraliser structure conjecture for formal maps

Let $K$ be a field of characteristic zero, let $\mathcal{G}$ be the group of formal maps under composition, and let $g\in\mathcal{G}$ be tangent to but not equal to the identity. For $h\in\mathcal{G}$, write $\{h^{\circ a}:a\in K\}$ for the one-parameter group of iterates of $h$. Centraliser structure conjecture. The centraliser $C_{\mathcal{G}}(g)$ is abelian and is the inner direct product of its torsion subgroup and a finite number of one-parameter groups of iterates $\{h^{\circ a}:a\in K\}$. Generically, the centraliser is just $\{g^{\circ a}:a\in K\}$, and the occurrence of a two-parameter subgroup corresponds to the possibility of conjugating $g$ to a product series. The conjecture holds in dimension $d=1$, while the general structure in characteristic zero remains open.