The reduced piercing conjecture for projected rectangle families

Let $\mathcal{F}$ be a finite family of axis-parallel rectangles, and let $\overline{\mathcal{F}}$ be the associated family obtained by intersecting each rectangle with a fixed collection of horizontal lines as in the source construction. Write $\tau$ and $\nu$ for piercing and matching numbers.

Projected-family piercing conjecture. There exists a constant $A$ such that every finite family of axis-parallel rectangles satisfies

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

This is presented as one of two conjectures that would imply a constant-factor bound in Wegner's conjecture. Its status is not resolved in the supplied context.