Equality of the quantum Galois deformation group in the first case

Let $q$ be the parameter occurring in the $q$-SI $\sigma$-differential setting, and let $\mathcal{NCF}_{L/k}(A)$ be the commutative deformation functor from Proposition 5.24i. For a commutative $L^{\natural}$-algebra $A$, write

$$\widehat{G}_{II}(A)=\{(e,b(W_1))\in A\times A[[W_1]]\mid e-1\text{ and all coefficients of }b(W_1)\text{ are nilpotent}\}.$$

Equality conjecture. If $q$ is not a root of unity, the inclusion $\mathcal{NCF}_{L/k}(A)\hookrightarrow\widehat{G}_{II}(A)$ is an equality.

This predicts that the explicitly described formal group captures all commutative deformations in this case. The source provides the inclusion but gives no resolution of the proposed equality.