This Demonstration shows that Cowley's net can be folded into a nonconvex solid with nonplanar faces.
In [1, pp. 2–3, 3], it was shown that Cowley's dodecarhombus net did not consist of golden rhombuses nor of rhombuses of rhombic dodecahedron, so it cannot be folded into a convex polyhedron. But if we consider Cowley's rhombuses as hinged equilateral triangles, his net can be folded into a nonconvex polyhedron. So in this case, rhombuses are a kind of skeleton in the sense of [4, p. 282], although not all dihedral angles are congruent.
According to [6, 7] the shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus  and is named after Stanko Bilinski, who rediscovered it in 1960. But the rhombuses of Cowley are 60° degree , so the previous statement is not true.