This is a follow-up to the question asked in Higher integrability for Sobolev functions

Updated question: Given the very helpful counterexamples and the ideas, I have the following question: Suppose for each $r>0$, there holds $$\int_{B_r(x)} |\nabla u|^2 dy \leq \frac{r}{c}\int_{\partial B_r(x)} |\nabla u|^2 d\sigma$$ for some $c \in (0,2)$ which is a uniform constant, then does this imply some form of higher integrability?

Clearly, when $c=2$, it is easy to see that $u \in W^{1,\infty}_{loc}(\mathbb{R}^2)$, so does $c<2$ still retain some form of higher integrability?