The completion conjecture for the $c_2$-invariant of primitive $\phi^4$ graphs

Let $G_1$ and $G_2$ be primitive $\phi^4$ graphs, and let $c_2^{(q)}(G)$ denote the $c_2$-invariant modulo a prime power $q$. Let $\mathcal{P}(G)$ denote the Feynman period. Completion conjecture.

$$\mathcal{P}(G_1)=\mathcal{P}(G_2)\quad\Longrightarrow\quad c_2^{(q)}(G_1)\equiv c_2^{(q)}(G_2)\pmod q$$

for all prime powers $q$. The paper proves the corresponding completion statement over prime fields using the Martin invariant, while the prime-power formulation remains open in the supplied text.