Concrete uncountable Moore–Schmidt conjecture

Let $\Gamma$ be a discrete group acting concretely on a measure space $X=(X,\mathcal{X},\mu)$, and let $K$ be a compact Hausdorff abelian group. A concrete $K_{\mathcal{B}a}$-valued cocycle on $X$ is a family $\rho=(\rho_\gamma)_{\gamma\in\Gamma}$. Concrete uncountable Moore–Schmidt conjecture. The cocycle $\rho$ is a concrete coboundary if and only if the $\mathbb{T}$-valued concrete cocycles $\hat{k}\circ\rho=(\hat{k}\circ\rho_\gamma)_{\gamma\in\Gamma}$ are concrete coboundaries for all $\hat{k}\in\hat K$. The only-if direction is easy; the if direction is the unresolved part, arising from the gap between abstract and concrete measurable maps.