Model originally uploaded to Thingiverse at https://www.thingiverse.com/thing:1271779.
Meanwhile, measure the inside of a shelf compartment, remove about 10mm on each measure: width, depth and height.
With wire (e.g. recycled wire hangers), cut 4 x height, 2 x width and 2 x depth.
Using cardboard (or any other similar material cats like to sleep on), make a bottom plate (width x depth).
Once the print is done, you should have 8 corners: 4 bottom clips and 4 top connectors.
Push the wire in the clips and the connectors as depicted in the other stl (not to be printed, it is just a model of how everything fits together).
Cover with an old t-shirt. Secure with safety pins or paper clips. Let him enjoy...
(No cat was harmed in the making of this box. Fingers, however, were. Cats are extremely territorial...)