Motivic realization conjecture for the Grothendieck–Teichmüller group

Let $K_*=(K_n)_n$ be the tower of motivic groups arising from the moduli spaces $\mathcal{M}_{0,n}$, and let $\mathcal{R}_{\mathrm{Betti}}(K_*)$ and $\mathcal{R}_{\mathrm{De\ Rham}}(K_*)$ denote its Betti and de Rham realizations. The weight filtration on the $K_n$ induces a weight filtration on the motivic Grothendieck–Teichmüller group. Motivic realization conjecture. There are isomorphisms

$$GT^{\mathrm{mot}}(\operatorname{Spec}(\mathbb{F}))\simeq GT\simeq\operatorname{Aut}(\mathcal{R}_{\mathrm{Betti}}(K_*)),$$

and, for the sum of the induced graded pieces denoted by $GRT^{\mathrm{mot}}(\operatorname{Spec}(\mathbb{F}))$,

$$GRT^{\mathrm{mot}}(\operatorname{Spec}(\mathbb{F}))\simeq GRT\simeq\operatorname{Aut}(\mathcal{R}_{\mathrm{De\ Rham}}(K_*))$$

The claim proposes that the motivic and classical Grothendieck–Teichmüller groups, and their graded versions, are recovered as automorphism groups of the corresponding realizations of the $K_n$ tower; the source presents this as an expectation and gives no resolution.