The Deligne–Ihara-type kernel conjecture for the thrice-punctured projective line

Let $a$ be the tangential base point $(0,\partial/\partial t_{|0})$ for $X:=\mathbb{P}^1_{\mathbb{Q}}\setminus\{0,1,\infty\}$. Let $G_{BK}(\mathbb{Q})$ be the relevant Bloch–Kato motivic fundamental group, let $K$ be the kernel of the map from the motivic fundamental group of $X$ to that of $\mathbb{Q}$, and let $\mathcal{I}\subset\operatorname{Lie}(G_{BK}(\mathbb{Q}))$ be the ideal generated by the degree $-1$ homogeneous elements. Deligne–Ihara-type kernel conjecture. The map

$$d\rho_a:\operatorname{Lie}(G_{BK}(\mathbb{Q}))\to\operatorname{End}(\operatorname{Lie}(K))$$

has kernel equal to $\mathcal{I}$. This is proposed as an analogue of the Deligne–Ihara conjecture and is intended to identify the special property of the representations arising from geometric sections. Its resolution is not stated in the supplied text.