Linear independence of indefinite iterated Eisenstein integrals

We prove linear independence of indefinite iterated Eisenstein integrals over the fraction field of the ring of formal power series $\mathbb{Z}[[q]]$. Our proof relies on a general criterium for linear independence of iterated integrals, which has been established by Deneufch{\^a}tel, Duchamp, Minh and Solomon. As a corollary, we obtain $\mathbb{C}$-linear independence of indefinite iterated Eisenstein integrals, which has applications to the study of elliptic multiple zeta values, as defined by Enriquez.