For tiling simply connected regions with shapes beyond just rectangles, is there a lower # of tile shapes needed for NP-completeness?
Found their paper on ArXiv. It is quite easy to go from coloured Wang tiles to uncoloured shapes by introducing zig-zag boundaries, like a jig-saw, as illustrated in their Figure 3. So then still 23 tiles are needed, as colours are simply replaced by boundaries. 

Their contribution is to move from boundaries to only rectangular tiles. The authors state in Lemma 3.3 how many rectangles are needed to replace coloured Wang tiles, and a long Section 5 is devoted to the proof of that lemma.