I am trying to show an estimate of the following form: Given any $p(x)$ such that $1<p^-\leq p(x) \leq p^+ <\infty$ and $p(\cdot)$ is log-Holder continuous, does there exists an $R_0$ (depending only on $p(\cdot)$, $n$ and log-Holder continuity of $p(x)$) such that
$$ \int_{B_R} M_{<{2R}} (|f|)^{p(x)}(x) \ dx \leq C \int_{B_{2R}} |f(x)|^{p(x)} \ dx + 1$$
holds for every function $f \in L^{p(\cdot)} (B_{2R})$ and every $R<R_0$.

Here $M_{<T}(|f|)(x) := \sup_{r<T} \frac{1}{|B(x,r)|}\int_{B(x,r)} |f(y)| \ dy$

EDIT: I have been informed by Peter Hasto that the above estimate is true only under a size restriction on $\int_{B_{2R}}|f(x)|^{p(x)} \ dx$.

EDIT: The answer to this question can be found in Theorem 4.8 and Corollary 4.9 of  arxiv.org/abs/1707.02535.