Equality of the absolute Galois group with the dessin automorphism subgroup

Let $\mathcal{L}$ be the lattice of biased dessins, let $\Gamma' = \operatorname{Aut}(\mathcal{L})$, and let $\Gamma'_0$ be the subgroup of $\Gamma'$ that fixes the three vertices of the bottom element $D_1$. The absolute Galois group $\Gamma$ is a subgroup of $\Gamma'_0$. Equality conjecture.

$$\Gamma = \Gamma'_0.$$

The preceding theorem establishes the inclusion, while the source states that there is currently no evidence or proof strategy for the reverse inclusion.