Pak–Yang's three-rectangle NP-completeness conjecture

A rectangle tiling set is a finite set $R$ of rectangles, and a simply connected region is a region in the plane without holes that is to be tiled using translated copies of rectangles from $R$. Pak–Yang's conjecture. There exists a set $R$ of $3$ rectangles such that tiling simply connected regions with $R$ is \textsf{NP}-complete. This conjecture concerns the minimum size of a fixed rectangle set yielding computationally intractable tiling; the paper proves the corresponding result for a set of $111$ rectangles, while the three-rectangle case remains open.