Determine the undetermined expansion coefficients for n greater than or equal to 5

Determine the expansion coefficients c_{\sigma_1\sigma_2\dots\sigma_k} in the iterated-integral expansion of d_n(g) that remain undetermined by the available sum rules when n\geq 5.

Background

The paper introduces universal coefficients c_{\sigma_1\sigma_2\dots\sigma_k} in the expansion of d_n(g) as a linear combination of iterated Chen integrals. By specializing the auxiliary functions f_+ and f_- and expanding the resulting identities around z=1/2, the authors derive sum rules for these coefficients. They report that these sum rules determine all coefficients for n\leq 4 but fail to determine some coefficients starting at n=5, leaving the general coefficient system unresolved for higher n.