Drinfel'd–Grothendieck conjecture on the profinite Grothendieck–Teichmüller tower

Let $\widehat{\operatorname{GT}}$ be the profinite Grothendieck–Teichmüller group, defined using the first two levels of the genus-zero stage of the profinite Grothendieck–Teichmüller tower, and let $\hat{\mathfrak T}^{\mathrm{out}}$ denote that full tower with outer continuous homomorphisms as maps. Drinfel'd–Grothendieck conjecture. The natural action of $\widehat{\operatorname{GT}}$ on the full tower induces an isomorphism

$$\widehat{\operatorname{GT}}\cong\operatorname{Aut}(\hat{\mathfrak T}^{\mathrm{out}}).$$

Drinfel'd, following Grothendieck's Esquisse d'un Programme, speculated that the group defined from the first two genus-zero levels acts on the full tower in this way. The supplied text gives no resolution status.