This is the 3D (and higher D) version of A claim on partitioning a convex planar region into congruent pieces

• Is there a 3D convex polyhedral solid that can be cut into 2 mutually congruent non-convex polyhedral solids but not into 2 mutually congruent convex solids? An example with say, 3 replacing 2 will also do.

Prof. Noam Elkies's construction on the above linked page (for planar case of the question) does not appear to readily yield a 3D answer.

Note: As has been noted more recently in the above linked page, even on the plane, I don't know any convex polygon with even number of sides that has the desired property.