Not that it matters because the four fixed corner assumption is physically impossible anyway, so there is no "in practice" to worry about. This can't happen.

@J... in principle we could have the four planets connected to each other by light rigid braces. If the planets are far enough apart the forces between them ought to be small enough that making the braces light enough is possible whilst maintaining sufficient rigidity. The actual limitation here would be getting enough material to produce the braces, as well as actually constructing them. It would also require that the planets have no individual rotation, but instead the entire assembly rotates as a whole (which may mess up the orbit, I'm not sure)

@Tristan Space is vast. I don't think you've considered the numbers. Fantasy will need to be invoked at some point to make this plan work. "Rigid" is not possible over such distances.

space is vast, but the forces between the bodies drop off fairly quickly. If the planets are far enough apart to barely be felt then the necessary station-keeping force from the brace is trivial. The braces could therefore have a tapering cross-section, being wide at the base mostly to support its own weight. This weight could also be used to provide some of the centripetal force needed for a rotating assemblage, allowing you to also cut the cross-section needed (similar to space elevators using the weight of the cable to keep the top in place)

you'd need an extremely empty region of space, and a loooooooot of material, and the ability to construct this assemblage on such a large scale, but in principle those are all engineering challenges not ones of physics, so it is not correct to say that the four fixed corner assumption is physically impossible

@Tristan "to provide some of the centripetal force needed for a rotating assemblage" - it can't be allowed to rotate or the squircle orbit becomes impossible.

are you sure rotating makes the orbit impossible? The exact details with obviously differ, but the same logic gone through in the answer itself ought to apply just as well to the rotating case

and obviously the situation wouldn't be able to last forever, this entire thing is dealing with unstable orbits to begin with, but given a light orbiting body, and the whole thing being distant from anything else then the only forces we need to concern ourselves with for the assemblage are from each massive object to the centre of the assemblage. This means our braces only need to resist compressive forces, and we don't have to concern ourselves with any bending forces

by moving the heavy bodies far apart, the compressive forces we need become as small as we like

so the only place for physics to come in here, is if it's impossible to build a sufficiently tall tower to reach from the massive body, to the centre of the assemblage

which definitely seems like an engineering problem, not a physics one

No bending forces? What do you think makes the planet turn around the corner? Have you calculated what torques look like on lever arms measured in astronomical units? Go do the numbers - don't trust me. It is impossible.