For tiling simply connected regions with shapes beyond just rectangles, is there a lower # of tile shapes needed for NP-completeness? In "TILING SIMPLY CONNECTED REGIONS WITH RECTANGLES" by Igor Pak and Jed Yang, they show there is a set of "no more than $10^6$ rectangles" such that the problem of tiling an arbitrary simply connected region with these rectangle shapes is NP-complete. However the number of rectangle shapes $10^6$ seems pretty extreme. Maybe it's hard to close the gap on P vs. NP-complete for tiling in terms of the number of rectangle shapes available, but what if we allow more complex pieces such as non-rectangular Tetris pieces and larger non-rectangular pieces? (We could insist the piece shapes are all simply connected, but I'm not adamant about that). 

Can we show NP-completeness for tiling a simply connected region with a much smaller set of (potentially non-rectangular) pieces available, compared to $10^6$?

The authors mention and rely upon an NP-completeness result for tiling with "Generalized Wang Tiles" that seem to use coloring as well as shape, and claim that NP-completeness can be achieved with a fixed set of 23 generalized Wang tiles. However I wasn't able to decipher this result to figure out how many ordinary non-colored (and e.g., perhaps also simply connected) tile shapes are needed to establish 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.