Panzer–Schnetz coaction conjecture for $\phi^4$-periods

Let $\mathcal{P}_{\phi^4}^{\rm m}$ be the space of motivic $\phi^4$-periods, let $\mathcal{P}^{\rm dR}$ be the de Rham period algebra, and let $\Delta$ denote the Galois coaction.

$$\Delta\colon \mathcal{P}_{\phi^4}^{\rm m}\longrightarrow \mathcal{P}^{\rm dR}\otimes\mathcal{P}^{\rm m}.$$

Panzer–Schnetz coaction conjecture. The Galois coaction closes on $\phi^4$-periods:

$$\Delta\big(\mathcal{P}_{\phi^4}^{\rm m}\big)\subseteq\mathcal{P}^{\rm dR}\otimes\mathcal{P}_{\phi^4}^{\rm m}.$$

The conjecture is supported by computations through loop order seven and by higher-loop examples, including periods that are polylogarithms at second and sixth roots of unity.