Brown's coaction conjecture for generalised scalar Feynman integrals

Let $\mathcal{P}_{g}^{\rm m}$ be the finite-dimensional $\mathbb{Q}$-vector space of motivic realisations of regularised generalised Feynman integrals associated to scalar Feynman graphs, and let $\Delta$ denote the Galois coaction.

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

Brown's coaction conjecture. The space $\mathcal{P}_{g}^{\rm m}$ is stable under the Galois coaction:

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

This extends the coaction conjecture from $\phi^4$-periods to regularised generalised Feynman integrals associated to arbitrary scalar Feynman graphs. Low-loop computations support the analogous $\phi^4$ statement.