The refined weight-mixing conjecture for $c_2=0$ $\phi^4$ graphs

Let $G$ be a primitive log-divergent $\phi^4$ graph with loop number $h_G$ and $c_2(G)=0$. Suppose $P(G)$ is mixed Tate. Refined weight-mixing conjecture. Either $P(G)$ has pure weight $2h_G-4$, or its weights include some weights between $h_G+2$ and $2h_G-5$.

The source says that the known $\phi^4$ data is consistent with this sharper alternative, while noting that the analogous statement is false for non-$\phi^4$ graphs. It remains open.