The non-degenerated conjecture for the depth-graded motivic Lie algebra

Let $\mathfrak{dg}_1$ be the depth-one component of the depth-graded motivic Lie algebra, let $\mathbb{P}$ be the space of restricted even period polynomials, and let $\mathrm{Lie}_n(\mathfrak{dg}_1)$ denote the component with exactly $n$ occurrences of the formal Lie bracket. For $n\geq 2$, define $\alpha$ by sending a period-polynomial relation tensored with $n-2$ elements of $\mathfrak{dg}_1$ to the corresponding iterated formal brackets, and let $\beta$ replace the formal Lie bracket by the induced Ihara bracket. Non-degenerated conjecture. For $n\geq2$, the sequence

$$\mathbb{P} \otimes\underbrace{\mathfrak{dg}_1\otimes\cdots\otimes\mathfrak{dg}_1}_{n-2}\xrightarrow{\alpha}\mathrm{Lie}_n(\mathfrak{dg}_1)\xrightarrow{\beta} \mathfrak{dg}_n$$

is exact. This conjecture predicts that the period-polynomial relations give all relations among the depth-one generators of the depth-graded motivic Lie algebra. The supplied text gives no resolution status.