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on Centered Hardy-Littlewood maximal constant in dimension $2$ ($C_{47}$)

The statement

In $\mathbb{R}^d$ ( $d\ge 1$ ), let $M_d$ denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by

M_d f(x)\ :=\ \sup_{r>0}\ \frac{1}{\lvert Q(x,r)\rvert}\int_{Q(x,r)} \lvert f(y)\rvert\,dy,

where $Q(x,r)$ is a closed $\ell_\infty$ ball of radius $r$ and center $x$ in $\mathbb{R}^d$ , that is, a closed cube centered at $x$ , with sides parallel to the coordinate axes, and sidelength $2r$ , and $\lvert\cdot\rvert$ denotes Lebesgue measure. [Ald2011-def-Md] [Ald2011-def-Qxr]

Let $c_d$ be the smallest constant such that for every $f\in L^1(\mathbb{R}^d)$ and every $\alpha>0$ ,

\alpha\,\bigl\lvert\{x\in\mathbb{R}^d:\ M_d f(x)\ge \alpha\}\bigr\rvert\ \le\ c_d\,\lVert f\rVert_1.

[Ald2011-def-cd]

We define

C_{47}\ :=\ c_2,

the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maximal operator associated to axis-parallel squares in $\mathbb{R}^2$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.