Equality for commutative deformations of the logarithmic q-difference extension

Let $q$ be the parameter in the $q$-SI $\sigma$-differential extension $(\mathbb{C}(t,\log t),\sigma,\theta^*)/\mathbb{C}$, and let $\mathcal{F}_{L/k}(A)$ be the commutative deformation functor for a commutative $L^{\natural}$-algebra $A$. Let

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

Equality conjecture. If $q$ is not a root of unity, the injection $\mathcal{F}_{L/k}(A)\rightarrow\widehat{G}_{III}$ is an equality.

This would show that the explicitly defined formal group accounts for all commutative deformations of the logarithmic extension. The source proves the injection but does not resolve the equality.