The $c_2=-z_2$ maximum-weight MZV conjecture

Let $G$ be a primitive log-divergent graph with loop number $h_G$ and $c_2(G)=-z_2$. $c_2=-z_2$ conjecture.

$$P(G)\in\mathcal{Z}_2,\qquad \operatorname{wt}(P(G))=2h_G-3.$$

Here $\mathcal{Z}_2$ is the algebra of multiple polylogarithmic periods at second roots of unity, as used in the source. The claim remains open.