A Chain Of Dynamic Cubes, Moving About Under Physics

I’ve been trying to get this working for a couple of days, tearing my hair out. I want to link a chain of two or more cubes to a master cube that is animated, which will move around chained together dynamically. The problem is that when the movement becomes sufficiently fast, the cube constraints (I am using “point”) doesn’t respect the distances and they start stretching like springs. I need them to stay exactly the same length. I cant’ find a way to achieve this; ramping up the solver steps and iterations doesn’t seem to help much. I’m at my wits end. Here’s a blend file-

SpringCubes0100.blend (641 KB)

The early frames of the animation show it basically working, but note that up around frame 316 the mauve cube has stretched a long way from the blue cube, as if the constraint has turned into a spring. Is there another way to achieve this, or some setting, or something? All I want is for the mauve-blue cube distance, and the red-mauve cube distance, to remain constant.