It is well-known that for $f \in L^1(\mathbb{R^n})$,$\mu(x \in \mathbb{R^n} | Mf(x) > \lambda) \le \frac{C_n}{\lambda} \int_{\mathbb{R^n}} |f| \mathrm{d\mu}$, where $C_n$ is a constant only depends on $n$.

It is easy to see $C_n \le 2^n$, but how to determine its optimal asymptotic behavior? For example, does $C_n$ bounded in $n$? Is $C_n$ bounded by polynomial in $n$?