The Hepp-bound faithfulness conjecture for primitive phi-four graphs

Let $G$ be a graph that is p-logarithmic in $d=4$ dimensions and whose vertices all have degree at most $4$; call such a graph a $\phi^4$ graph. Let $\mathcal{P}(G)$ denote its period and $\mathcal{H}(G)$ its Hepp bound. Hepp-bound faithfulness conjecture. Two $\phi^4$ graphs have equal periods if and only if they have equal Hepp bounds. The conjecture is supported by computations for primitive $\phi^4$ graphs through $11$ loops, but remains open; it is motivated by analogous conjectures for the $c_2$ invariant and the permanent.