Very interesting! I believe it could be made by shapemodding a 4x4x4. Each "1/4th cone" on the top and bottom would be the equivalent of a single center piece, a wing edge from each of two neighboring faces, and a corner. The center pieces would be a little more difficult, but you would have to somehow split the edges.
ConeCube.png [ 16.13 KiB | Viewed 959 times ]
Green is what would become the "1/4th cone" pieces.
Red would be one center piece and blue would be another.
If a SQ-1 core had an additional cut through the core perpendicular to the existing one I think that would be possible by changing the shape of the pieces.
EDIT: Tried making a physical version of this by splitting the edges. That worked fine, but face turns can't be made when the horizontal slice layers are in a 45° turn because the center pieces cannot rotate at that place in the core. Even if they could, I noticed the issue of the core not turning with slice layers at times, meaning the track for a vertical 180° move would be blocked. Seems like it would be a pretty simple shapeways build with a ball core though. Just modifications to centers and edges and some bandaging.