Folklore conjecture on the algebra of multizeta values in positive characteristic

Let $\overline{\mathfrak{Z}}$ be the $\overline{\mathbb{Q}}$-algebra generated by multizeta values, and let $\overline{\mathfrak{Z}}_w$ be the $\overline{\mathbb{Q}}$-vector space spanned by the multizeta values of weight $w$, for $w\geq 2$. Write $\mathfrak{Z}$ for the corresponding $\mathbb{Q}$-algebra generated by multizeta values. Folklore conjecture. The algebra $\overline{\mathfrak{Z}}$ is weight-graded,

$$\overline{\mathfrak{Z}}=\overline{\mathbb{Q}}\oplus\bigoplus_{w\geq 2}\overline{\mathfrak{Z}}_w,$$

and is defined over $\mathbb{Q}$, meaning that the canonical map

$$\overline{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathfrak{Z}\longrightarrow\overline{\mathfrak{Z}}$$

is bijective. This is presented as a folklore open problem concerning the algebraic structure and field of definition of multizeta values; the supplied text gives no resolution.