Wegner's piercing conjecture for axis-parallel rectangles

For a finite family $\mathcal{F}$ of axis-parallel rectangles in $\mathbb{R}^2$, let $\tau(\mathcal{F})$ denote its piercing number and $\nu(\mathcal{F})$ its matching number.

Wegner's conjecture. Every finite family of axis-parallel rectangles in $\mathbb{R}^2$ satisfies

$$\tau(\mathcal{F})\leqslant 2\nu(\mathcal{F}).$$

The conjecture is open. The weaker assertion that $\tau(\mathcal{F})\leqslant c\nu(\mathcal{F})$ for an absolute constant $c$ is also open; the best supplied upper bound is $O(\nu)(\log\log\nu)^2$.