Existence of a motivic Galois pro-NGLA realizing the Hom conjecture

Let $K$ be a number field and let $X$ and $X'$ be smooth hyperbolic curves defined over $K$. Denote by $\pi_1^{mot}(\bar X)$ the motivic fundamental group over $\bar K$, and let $\mathcal{G}al^{mot}(\bar K/K)$ be a motivic Galois pro-NGLA. Motivic Galois realization conjecture. There exists a motivic Galois pro-NGLA $\mathcal{G}al^{mot}(\bar K/K)$ which canonically maps to $\pi_1^{mot}(\bar X)$ for every smooth hyperbolic curve $X$ over $K$, such that

$$\operatorname{Hom}_{K}(X,X')\longrightarrow \operatorname{Hom}_{\mathcal{G}al^{mot}(\bar K/K)}(\pi_1^{mot}(\bar X),\pi_1^{mot}(\bar X'))$$

is a natural one-to-one correspondence. This conjecture proposes a motivic Galois object whose action makes the NGLA version of the preceding anabelian correspondence true; no resolution is given in the paper.