Algebraic freeness of mock Eisenstein and classical Eisenstein generators

Determine whether the algebra generated by the mock Eisenstein series f_2, f_4, f_6, … together with the Eisenstein series G_2, G_4, and G_6 is free, equivalently whether these functions satisfy no algebraic relations.

Background

Theorem 1.2 introduces the family of mock Eisenstein series f_k and defines the algebra F := Q[f_2, f_4, …, G_2, G_4, …]. The paper establishes that F is closed under differentiation, while Remark 1.7 conjecturally anticipates that F is freely generated by the even-indexed f_k together with G_2, G_4, and G_6.

Section 6 reports that the absence of algebraic relations among f_2, f_4, f_6, …, G_2, G_4, and G_6 has been verified numerically through weight 24, and also through mixed weight 12. The unresolved question is whether this apparent algebraic independence holds in general, so that the algebra F is free.