Origin of relations beyond shuffle antipode for multiple Eisenstein series

Determine whether all independent relations among multiple Eisenstein series predicted by the above dimension conjecture but not obtained from shuffle antipode relations are generated by differentiating with respect to \tau, i.e., ascertain whether the difference between the predicted number of relations (\#rel_conj) and the number derived from shuffle antipode relations and their products (\#rel_anti) is entirely explained by lifting relations in \mathcal{E}_k to \mathcal{E}_{k+2} via \tau-derivation.

Background

The authors compare the number of relations predicted by the dimension conjecture with those deduced from shuffle antipode relations, finding a gap in certain weights. They suggest a mechanism—differentiation with respect to \tau—that could lift relations to higher weights, potentially accounting for the gap.

Confirming this would both validate the dimension conjecture’s counts and provide a systematic construction of ‘missing’ relations across weights, clarifying the interplay between analytic operations and algebraic structures in the theory of multiple Eisenstein series.