I have to look into that linkage stuff myself, it’s great to simplifiy the math.
As far as i remember linkages it’s all about the pivot points, so my first guess is that a sprocket will make things more complicated. The sprocket radius just would add more math. I didn’t realise that in my initial idea either.

Okay, I’ve given it some thought. You could put fixed sprockets on the end of one of the triangular geometry designs, as long as the action line of the sprocket is aligned with the action line of the arm. You would need to do something to be sure the chain that goes between the arms isn’t affecting their rotation. For example, you could have the chain hang loose between them and/or replace that section of chain with something lightweight like twine. In addition, you would want to mark the chain links that engage the sprocket so you are hanging the sled the same way every time. Not sure it would be worth the effort, but it should be doable.

could you please diagram this, I’m not visualizing what you are talking about.

Something like this:

The chains form a vee at the bottom because I was too lazy to lay out a sagging arc. If I were to do this (and again, I’m not convinced it’s worth the effort; I’m just exploring @vertex’s idea), I would make the sagging vee part out of lightweight material so it doesn’t affect the trajectory of the pantograph arms.

I think you could skip the sprockets and make somekind of springloaded quick locking clamp to grab the chain on the linkages.

though the sprockets in this position are interesting