Conjectural algebra and modular correspondence for multiple Eisenstein series

Let $\rho^*$ and $\psi^*$ be products of representations of the form $\rho^*_{U_i}$, and let $\mathcal{W}^{\rho^*}_n$ denote the $\mathbb{F}_p$-vector space of multiple Eisenstein series of degree $\binom{\rho^*}{n}$. Write $\rho^*=\otimes_j(\rho^*_{t_j})^{\otimes\nu_j}$ and $\sigma=\prod_j\chi_{t_j}^{\nu_j}$. For the relevant composition arrays $\mathcal{C}$ and $\widehat{\mathcal{C}}^*$, let $\mathcal{Z}^{\sigma}_n$ be the space of multiple zeta values of degree $\binom{\sigma}{n}$ and let $\mathcal{E}_A(\widehat{\mathcal{C}}^*)$ be the associated multiple Eisenstein series.

Multiple Eisenstein correspondence conjecture. The following properties hold: (1) $\mathcal{W}^{\rho^*}_m\otimes\mathcal{W}^{\psi^*}_n\subset\mathcal{W}^{\rho^*\otimes\psi^*}_{m+n}$; (2) the correspondence $\zeta_A(\mathcal{C})\mapsto\mathcal{E}_A(\widehat{\mathcal{C}}^*)$ defines an isomorphism $\eta$ between $\mathcal{Z}^{\sigma}_n$ and $\mathcal{W}^{\rho^*}_n$, compatible with multiplication so that $\mathcal{W}:=\sum_{n,\rho^*}\mathcal{W}^{\rho^*}_n$ is a graded $\mathbb{F}_p$-algebra under $\otimes$, isomorphic to $\bigoplus_{n,\sigma}\mathcal{Z}^{\sigma}_n$; and (3) $f\in\mathcal{Z}^{\sigma}_n$ is eulerian if and only if $\eta(f)$ is a modular form in $M_n(\rho^*;\mathbb{L}_\Sigma)$.

The conjecture proposes a multiplicative and graded correspondence between multiple zeta values and multiple Eisenstein series, including a characterization of eulerian elements by modularity. The source gives no resolution, so it remains open.