The formal multiple Eisenstein series spanning conjecture

Let $\mathcal{G}^{\!f}$ be the commutative $\mathbb{Q}$-algebra of formal multiple Eisenstein series, with elements $G^{\!f}(k_1,\dots,k_r)$ and double-indexed elements $G^{\!f}\bi{k_1,\dots,k_r}{d_1,\dots,d_r}$. Let $\operatorname{Fil}^{\mathrm{lwt}}_0\mathcal{G}^{\!f}$ be the subspace spanned by the single-indexed elements $G^{\!f}(k_1,\dots,k_r)$. Formal multiple Eisenstein series spanning conjecture. The map $G^{\!f}:(\mathfrak H^1,\ast)\to\mathcal{G}^{\!f}$ sending $z_{k_1}\cdots z_{k_r}$ to $G^{\!f}(k_1,\dots,k_r)$ is surjective; equivalently, $\mathcal{G}^{\!f}\simeq\operatorname{Fil}^{\mathrm{lwt}}_0\mathcal{G}^{\!f}$. This is presented as a formal version of conjectures attributed to Bachmann, Bachmann–Kühn, and Bachmann–Ihara; only special cases are known.