Positive measure of conjugation preimages of nonempty open sets

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable topological abelian group, both equipped with their Borel measurable structures. Let $D$ specify additive automorphisms $\rho_g$ of $A$ with $\rho_1=\mathrm{id}$, $\rho_{gh}=\rho_g\circ\rho_h$, and jointly continuous $(g,a)\mapsto\rho_g(a)$; a measurable function $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$; and an open neighborhood $N$ of $1$ on whose square $c$ is continuous. On $E=A\times G$, define $(a,g)\star(b,h)=(a+\rho_g(b)+c(g,h),gh)$ and $(a,g)^\dagger=(\rho_{g^{-1}}(-(a+c(g,g^{-1}))),g^{-1})$. Equip $E$ with the topology generated by all sets $\{z\star(a,g):a\in P,\ g\in Q\}$, where $z\in E$, $P$ is an open neighborhood of $0$ in $A$, $Q$ is an open neighborhood of $1$ in $G$, and $c$ is continuous on $Q\times Q$, and use its Borel measurable structure. Suppose $\mu$ is a nonzero measure on $E$, finite on every compact subset, and satisfies $\mu(\{z\star w:w\in B\})=\mu(B)$ for every $z\in E$ and every measurable $B\subseteq E$ with compact closure. For every nonempty open set $W\subseteq E$ and every $x\in E$, the set $\{y\in E:(x\star y)\star x^\dagger\in W\}$ has strictly positive $\mu$-measure. This conclusion permits that measure to be infinite.