The $c_2=-z_3$ period conjecture

Let $G$ be a primitive log-divergent graph with loop number $h_G$ and $c_2(G)=-z_3$. $c_2=-z_3$ conjecture.

$$P(G)\in \mathrm{i}\sqrt{3}\bigl(\mathcal{Z}_6\cap\mathrm{i}\mathbb{R}\bigr),\qquad \operatorname{wt}(P(G))=2h_G-3.$$

The three known examples lie in the indicated space, and the source observes that this would exclude ordinary MZVs. The all-graphs assertion remains open.