A new pyramid-like shape always lands the same side up

This is categorically different from the Gömböc, because it doesn't have uniform density. Most of its mass is concentrated in the base plate.

> This tetrahedron, which is mostly hollow and has a carefully calibrated center of mass

Uniform density isn't an issue for rigid bodies.

If you make sure the center of mass is in the same place, it will behave the same way.

If the constraints are that an object has to be of uniform density, convex, and not containing any voids, then you cannot choose where its centre of mass will be, other than by changing it shape.

That isn't true.

Look at the pictures. It has the same outer shape, that is all that is required for the geometry.

And for center of mass, you set the positions for the bars, any variations in their thickness, then size and place the flat facet, in order to achieve the same center of mass as for a filled uniform density object of the same geometry.

As the article says:

> carefully calibrated center of mass

Unless an object has internal interactions, for purposes of center of mass you can achieve the uniform-density-equivalent any way you want. It won't change the behavior.