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This is a nice question, and I look forward to seeing a definitive answer. Meanwhile, let me point out that it is known that a convex polytope in $E^n$ is not always determined by a finite set of its projections, but it is determined by all of its 2D projections. Richard Gardner, in Geometric Tomography, puts the first point this way: "it is generally not possible to choose a finite set of subspaces in such a way that the corresponding projections distinguish $P$ from every other convex polytope" [p.93].