Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open,
and which some might find intriguing.
Define an *equiprojective polyhedron* $P \subset \mathbb{}R^3$ as one whose
orthogonal projections—with
the exception of projections in directions parallel to a face—are
all $n$-gons for the same $n$. The definition is due to
Shephard, 45 years ago.
Thus, cubes are 6-equiprojective: their only projections to quadrilaterals are along directions parallel to faces. A triangular prism is 5-equiprojective.
There are no 3- or 4-equiprojective polyhedra, as established in the paper that
recently reopened this dormant subject:

Hasan, Masud, Mohammad Monoar Hossain, Alejandro López-Ortiz, Sabrina Nusrat, Saad Altaful Quader, and Nabila Rahman. "Some new equiprojective polyhedra." arXiv:1009.2252 (2010).

This paper establishes that both the equitruncated pyramid and the equitruncated
triangular cupola are 10-equiprojective:

I believe this remains open:

Q1. Is there a $k$-equiprojective polyhedron for every $k \ge 5$? If not, for which $k$ does there exist $k$-equiprojective polyhedra?

**Addendum**. After seeing Ian's nice resolution of

**Q1**, I went back to the cited paper and found this was already known: "(in fact, any $p$-gonal prism is $p + 2$-equiprojective)." My apologies! So I guess the real open problem here is:

Q2. Describe (or construct) all equiprojective polyhedra.