Three-rectangles NP-completeness conjecture for simply connected tilings

Determine whether there exists a fixed set R of exactly three rectangular polyomino tiles such that the decision problem of tiling arbitrary finite simply connected regions with translated copies of rectangles from R is NP-complete.

Background

This paper proves NP-completeness for tiling finite simply connected regions using 23 Wang tiles and, via a variant and a reduction, using 111 rectangles, improving previous bounds. The authors then highlight a long-standing conjecture originating from Pak and Yang concerning whether only three rectangles suffice to yield NP-completeness for simply connected tilings.

Resolving this conjecture would sharply reduce the number of rectangles required from 111 to 3, establishing a minimal fixed rectangular tile set that makes the simply connected tiling problem NP-complete.