Weak reverse Hölder inequalities for doubling measures

Let $μ$ be a doubling Radon measure in $\mathbb R^d$ with support $X$. We prove a local and quantitative form of the Bennewitz--Lewis theorem in this setting: if a Radon measure $ν$ satisfies the Bennewitz--Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, t