Injectivity of the formal-to-analytic multiple Eisenstein realization

Prove or disprove injectivity of the algebra homomorphism from the algebra of formal multiple Eisenstein series generated by admissible formal indices to the algebra of analytic multiple Eisenstein series, thereby determining whether the formal and analytic realizations are isomorphic.

Background

The paper introduces the realization map $\rho:E^{\mathrm f}\to E$ sending each formal multiple Eisenstein series to the corresponding analytic multiple Eisenstein series. Earlier work establishes that this map is a well-defined surjective algebra homomorphism.

The unresolved issue is whether distinct formal series can acquire the same analytic realization. The paper explicitly records that injectivity is unknown and does not address it; the present results instead establish the \mathfrak{sl}_2-structure on the analytic algebra.