Non-Abelian gauge-fixing conjecture for $L^p$ potentials

Let $P\to M$ be a smooth principal bundle, let $2p\geq\dim M$, and let $\operatorname{Ad}_*P$ and $\operatorname{Ad}P$ denote the adjoint coefficient bundles for potentials and gauge transformations. Suppose

$$\alpha\in L^p(\Lambda^1M\otimes\operatorname{Ad}_*P)$$

is a gauge potential with

$$F(\alpha)\in L^p(\Lambda^2M\otimes\operatorname{Ad}_*P).$$

Non-Abelian gauge-fixing conjecture. There exists a gauge transformation $\lambda\in W^{1,p}(\operatorname{Ad}P)$ such that

$$\alpha^\lambda:=\operatorname{Ad}_*(\lambda^{-1})\alpha+\lambda^{-1}d\lambda\in W^{1,p}(\Lambda^1M\otimes\operatorname{Ad}_*P).$$

The Abelian case is established in the source, while this conjectural extension would provide the gauge-fixing needed for general homogeneous spaces.