Dimension generating function for spaces of multiple Eisenstein series

Prove the conjectural formula for the generating function of the dimensions of the Q-linear spaces \mathcal{E}_k spanned by multiple Eisenstein series G_{k_1,\dots,k_r}(\tau) of total weight k, namely establish that \sum_{k\ge 0} \dim \mathcal{E}_k \, X^{k} = 1/(1 - X^2 - X^3 - X^4 - X^5 + X^8 + X^9 + X^{10} + X^{11} + X^{12}).

Background

The paper defines \mathcal{E}_k as the Q-linear span of multiple Eisenstein series of total weight k and discusses the expected number of independent relations among these series by weight. Building on conjectures of Okounkov and of Bachmann–van Ittersum–Matthes, the authors formulate a single generating function that predicts the dimensions of these spaces.

Establishing this formula would give a precise structural description of the graded dimensions of the algebra generated by multiple Eisenstein series, thereby clarifying the landscape of relations and bases across weights.