Of affine images of a rhombododecaedron circumscribed about a convex body in $\mathbb R^3$

The main result of the paper is dual to an earlier theorem by the author concerning affine images of a cubeoctahedron inscribed in a three-dimensional convex body. The rhombododecaedron is the polytope dual to the cubeoctahedron; the latter is the convex hull of the midpoints of the edges of a cube.
Theorem. Every convex body in $\mathbb R^3$ except for those mentioned below admits an affine-circumscribed rhombododecaedron. A possible exception is a body containing a parallelogram $P$ and contained in a cylinder over $P$.
The author does not know whether there is a three-dimensional convex body exceptional on the sense of the above theorem.