Gauge deformations for Hopf algebras with the dual Chevalley property

Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation $\zeta$ on $A$ such that A^{\zeta}\cong Q#H where $A^{\zeta }$ is the dual quasi-bialgebra obtained from $A$ by twisting its multiplication by $\zeta$, $Q$ is a connected dual quasi-bialgebra in ${}_H^H\mathcal{Y}\mathcal{D}$ and Q #H is a dual quasi-bialgebra called the bosonization of $Q$ by $H$.