The formal multiple Eisenstein subalgebra conjecture

Let $\mathcal{E}^{\!f}$ be the subspace of $\operatorname{Fil}^{\mathrm{lwt}}_0\mathcal{G}^{\!f}$ spanned by $G^{\!f}(k_1,\dots,k_r)$ with $r\geq0$ and $k_1,\dots,k_r\geq2$, and let $\mathcal{E}$ be the algebra spanned by the corresponding multiple Eisenstein series $\mathbb{G}_{k_1,\dots,k_r}$. Formal multiple Eisenstein subalgebra conjecture. The algebra $\mathcal{E}^{\!f}$ is an $\mathfrak{sl}_2$-subalgebra of $\mathcal{G}^{\!f}$, and the map

$$\mathcal{E}^{\!f}\longrightarrow\mathcal{E},\qquad G^{\!f}(k_1,\dots,k_r)\longmapsto\mathbb{G}_{k_1,\dots,k_r},$$

is an algebra isomorphism; in particular, $\mathcal{E}$ is also an $\mathfrak{sl}_2$-algebra. The conjecture would identify the formal and classical multiple Eisenstein subalgebras and establish the expected $\mathfrak{sl}_2$ structure; the source gives no resolution status.