Broadhurst's generalized parity conjecture for sixth-root multiple polylogarithms

Let $z_1,\ldots,z_d$ be sixth roots of unity, and let $\boldsymbol{n}=(n_1,\ldots,n_d)$ be a multi-index with weight $\left\lvert\boldsymbol{n}\right\rvert=n_1+\cdots+n_d$. For a multiple polylogarithm $\operatorname{Li}_{\boldsymbol{n}}(\boldsymbol{z})$, its depth is $d$.

Generalized parity conjecture. If $\left\lvert\boldsymbol{n}\right\rvert+d$ is odd, then the real part $\operatorname{Re}\operatorname{Li}_{\boldsymbol{n}}(\boldsymbol{z})$ has depth at most $d-1$. If $\left\lvert\boldsymbol{n}\right\rvert+d$ is even, then the imaginary part $\operatorname{Im}\operatorname{Li}_{\boldsymbol{n}}(\boldsymbol{z})$ has depth at most $d-1$.

This refines the parity theorem for alternating sums at sixth roots of unity; the source notes that the original formulation is stated modulo products and is equivalent by a recursive argument. The status of this sixth-root refinement is not resolved in the supplied text.