Reverse Hölder self-improvement conjecture

Let $w$ be a weight on cubes $Q$, and define the reverse Hölder characteristic by

$$[w]_{\operatorname{RH}_p}:=\sup_Q\frac{\left(|Q|^{-1}\int_Q w^p\,dx\right)^{1/p}}{|Q|^{-1}\int_Qw\,dx}.$$

Reverse Hölder self-improvement conjecture. If $w\in\operatorname{RH}_p$ and $1<p<\infty$, then for

$$0<\varepsilon\lesssim_{d,p}\frac{1}{[w]_{\operatorname{RH}_p}^{p}},$$

one has

$$\frac{1}{|Q|}\int_Qw^{p+\varepsilon}\,dx\le 2[w]_{\operatorname{RH}_p}^{p+\varepsilon}\left(\frac{1}{|Q|}\int_Qw\,dx\right)^{p+\varepsilon}.$$

This conjectures a quantitative higher-integrability estimate for reverse Hölder weights with an exponent range controlled by the reverse Hölder characteristic; the source gives no evidence that it has been resolved.