If you look at the attached pictures you can see designs for a #-bar / dr4b that allows for one motor to power the lift and the cap flipper. They will flip the cap when the lift is lifted, and then the lift can be lowered without the cap flipping back. This is for if you only have one motor left so you can't do an active flipper but you still want to be able to stack on the poles while easily flipping low caps. It would work best with a 4-bar because a 4-bar can be partially lifted while outside of the expansion zone.
Main problem with them is that the gear box is kind of big and hard to build, but on the one with no string you can put the motor on the top of the lift to eliminate some friction.
The angles I put in the picture are just for a general idea of scale, they don't actually mean anything
Or you can just build something like this:
Edit: If for some reason someone actually decides to build the designs in the image and they work can they share a picture with me because I can't make them work with a reasonable sized gear box, although I want to retry building them with Antichambers new tiny ratchet.