Let $u \in W^{1,p}(\mathbb{R}^n) \in L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following Maximal function $$ \mathcal{M}_{\leq k}(|\nabla u|)(x) = \sup_{R>0}\frac{1}{|\{B_R(x):u \leq k\}|} \int_{\{B_R(x):u \leq k\}} |\nabla u| \ dx. $$

Question: Is it true that $\mathcal{M}_{\leq k}(|\nabla u|)(x)$ is finite almost everywhere?

Analogous question regarding $\mathcal{M}_{\geq k}(|\nabla u|)(x)$ can also be asked.