Drinfeld's Grothendieck–Teichmüller conjecture for the motivic Galois group

Let $\mathcal{MTM}(\mathbb{Z})$ be the Tannakian category of mixed Tate motives over $\mathbb{Z}$, and let $\mathcal{G}_{1}=\mathbb{G}_{m}\ltimes\mathcal{U}_{1}$ be its motivic Galois group, with prounipotent part $\mathcal{U}_{1}$. The map $\lambda_{1}:\mathcal{U}_{1}\to\exp\mathfrak{t}_{3,1}^{0}$ identifies the motivic Galois group with a subgroup of the prounipotent Grothendieck–Teichmüller group $\mathrm{GRT}_{1}\subset\exp\mathfrak{t}_{3,1}^{0}$. Drinfeld's Grothendieck–Teichmüller conjecture.

$$\operatorname{im}(\lambda_{1})=\mathrm{GRT}_{1}.$$

The conjecture asserts that the motivic Galois action on the motivic fundamental torsor of paths on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ is governed exactly by the prounipotent Grothendieck–Teichmüller group. Brown proved that $\lambda_{1}$ is injective, but equality with $\mathrm{GRT}_{1}$ remains open.