This puzzle is part of the Monthly Topic Challenge #11: Now in 3D.
There are nine distinct known three-cell Garden of Eden patterns in the Life Without Death cellular automata, including the two well-known triominoes. The nine pieces are as follows:
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Note that some of these pieces are disconnected, but you can't dismantle them. The cells inside each piece must retain their relative positions.
The task is, can these nine pieces (treated as 1-cell high prisms) be fit perfectly into a 3x3x3 cube?