Quantum Galois group as the third quantum formal group

Let $q$ be the parameter of the $q$-SI $\sigma$-differential extension, let $\mathcal{NCF}_{L/k}$ be the noncommutative deformation functor, and let $\widehat{QG}_{III,q}$ be the quantum formal group receiving the functorial injection of Proposition 6.8o.

Quantum Galois group conjecture. If $q$ is not a root of unity, the injection in Proposition 6.8o is an equality, so

$$\mathcal{NCF}_{L/k}\simeq\widehat{QG}_{III}.$$

Consequently, the quantum Galois group of the $q$-SI $\sigma$-differential extension is the quantum formal group $\widehat{QG}_{III,q}$.

This identifies the quantum Galois group with the formal group constructed from the deformation theory. The preceding proposition gives only the injection, and the supplied text does not state that the conjectural equality is known.