Stabbing boxes with finitely many axis-parallel lines and flats

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in ${\mathbb{R}}^d$ to be pierceable by finitely many axis-parallel $k$-flats, where $0\le k<d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.