Reverse estimate of Hardy-Littlewood maximal function in Sobolev spaces

Question

Given $f\in L^1_{loc}(\mathbb{R}^n)$, we consider the maximal function $$ Mf(x) := \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|dy. $$ It is known that $M : W^{1,p}(\mathbb{R}^n)\to W^{1,p}(\mathbb{R}^n)$, for $1<p<\infty$, is a bounded operator : for any $f\in W^{1,p}(\mathbb{R}^n)$, it holds $$ \|Mf\|_{W^{1,p}} \leq C\|f \|_{W^{1,p}}, $$ for some $C>0$ independant of $f$. However, I did not find anything about the reverse estimate : $$ \|f\|_{W^{1,p}} \leq C\|Mf \|_{W^{1,p}}. $$ If we replace the $W^{1,p}$-norms by $L^p$-norms, then it is a consequence of the fact that Lebesgue points have full measure. However, it does not seem straightforward to me that an inequality of the form $|\nabla f| \leq C|\nabla Mf|$ holds in general.
Is there any known counterexample to this reverse estimate ? Or is this obviously true ?