A note on the limiting weak-type behavior for maximal operators

We study the following open question raised by Janakiraman in (2006): for $f\in L^1 (\mathbb{R}^n )\cap L^{\infty} (\mathbb{R}^n )$ and $\lambda > 0$, what is the limiting behavior of

$$\left[m\left(\{x \in \mathbb{R}^n :M(\vert f\vert^p )(x)>\lambda\}\right)\right]^{1/p}$$

as $p\to\infty$? In this note, we give a complete answer to this question.