The sl2-algebra conjecture for multiple Eisenstein series

Let $\mathcal{E}$ be the $\mathbb{Q}$-algebra generated by multiple Eisenstein series $G_{k_1,\ldots,k_r}(\tau)$ with $k_i\geq2$. Define maps on these generators by

$$W'(G_{k_1,\ldots,k_r})=(k_1+\cdots+k_r)G_{k_1,\ldots,k_r},$$ $$D'(G_{k_1,\ldots,k_r})=2\pi i\frac{d}{d\tau}G_{k_1,\ldots,k_r},$$

and

$$\delta'(G_{k_1,\ldots,k_r})= \begin{cases} -\frac12G_{k_2,\ldots,k_r},&k_1=2,\\ 0,&k_1>2. \end{cases}$$

The $\mathfrak{sl}_2$-algebra conjecture for multiple Eisenstein series. These maps should be well-defined $\mathbb{Q}$-linear maps on $\mathcal{E}$, should be derivations, and should form an $\mathfrak{sl}_2$-triple:

$$[W',D']=2D',\qquad[W',\delta']=-2\delta',\qquad[\delta',D']=W'.$$

Thus $\mathcal{E}$ should be an $\mathfrak{sl}_2$-algebra. This conjecture generalizes the known $\mathfrak{sl}_2$-structure on the algebra of quasi-modular forms. Its validity for the full algebra of multiple Eisenstein series remains open.