This is specifically about convex polyhedra, I don't see how rotating could help.
Note the egg shape in the article. specifically the widest band around the equator. now imagine one passing straight down through the other. one edge ring would pass through the shadow if it has a slight rotation offset but it is blocked by the next edge ring up, which could also fit but requires a different offset, so if you could change that rotational offset while it is passing through would it fit?
There are also other rotation profiles, hard to say if they would help or not.
example 1. a cube rifling faces parallel would not generate a concave cut volume but moving diagonally point to far point it would. really whenever a point sticks out. Unfortunately a cube already fits and any help a convex cut volume provides is not needed.