Borel fractional colorings of Schreier graphs

Let $\mathrm{\Gamma }$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $\mathrm{\Gamma }$ (with respect to some finite subset $F\subseteq \mathrm{\Gamma }$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $\mathrm{\Gamma }$ is the free group, answering a question of Meehan.