The coaction and weight conjectures for six-dimensional phi-three periods

Let $\mathcal{P}_{\phi^3}^{\mathfrak{m}}$ denote the space of motivic periods of six-dimensional $\phi^3$ theory, and let $\mathcal{P}^{\mathfrak{dr}}$ denote the de Rham period space. Write $\mathcal{P}_{\phi^3,\leq n}^{\mathfrak{m}}$ for the $\mathbb{Q}$-vector space of motivic $\phi^3$ periods at at most $n$ loops, and let $\Delta'$ be the non-trivial part of the motivic Galois coaction. Coaction and weight conjecture. The following assertions hold:

1. The number content of six-dimensional $\phi^3$ periods is a subset of, and possibly equal to, the number content of four-dimensional $\phi^4$ periods.
2. The motivic Galois coaction closes on $\phi^3$ periods:

$$\Delta\colon\mathcal{P}_{\phi^3}^{\mathfrak{m}}\longrightarrow\mathcal{P}^{\mathfrak{dr}}\otimes\mathcal{P}_{\phi^3}^{\mathfrak{m}}.$$

3. The motivic Galois coaction is consistent with the loop grading, with its non-trivial part provided by periods of lower loop order:

$$\Delta'\colon\mathcal{P}_{\phi^3,\leq n}^{\mathfrak{m}}\longrightarrow\mathcal{P}^{\mathfrak{dr}}\otimes\mathcal{P}_{\phi^3,\leq n-1}^{\mathfrak{m}}.$$

4. The maximum weight in $\mathcal{P}_{\phi^3,\leq n}^{\mathfrak{m}}$ is $2n-3$. These assertions organize the expected number-theoretical content, coaction structure, loop grading, and weight growth of six-dimensional $\phi^3$ periods. The source reports compatibility with the conjecture, while noting that the corresponding fourth assertion is proved in $\phi^4$ theory but is more obscure in $\phi^3$ theory; no full resolution is given.