Establish a constant guillotine-separation fraction for disjoint rectangles

Prove that every finite family of pairwise interior-disjoint axis-parallel rectangles contains a recursively guillotine-separable subfamily consisting of a constant fraction of the rectangles.

Background

The recursive guillotine cutting problem studies packings of pairwise interior-disjoint axis-parallel rectangles and asks how many rectangles can be preserved by recursively partitioning the bounding box with full horizontal or vertical cuts. The corresponding separable fraction is the proportion that can be retained by some guillotine cutting sequence.

For general disjoint rectangles, the paper states that Pach and Tardos proved only an Ω(1/log n) lower bound, where n is the number of rectangles, and conjectured that a constant fraction always suffices. The paper explicitly identifies this conjecture as unresolved. Such a result would also yield improved approximation guarantees for related independent-set problems.