I will throw this out there but it would be fairly invasive but should be very clean once finished. It would consume the least space underneath the bed:
Four large acme lead screw pillars that the bed frame rides on. You probably want those lead screws to run all the way to mountings on the ceiling for stability and safety unless these lead-screws are unreasonably.
You will still have the issue that the if driven independently the motors will not synch up without electronic controls so you can choose whether to install one or whether to gang up the four lead-screws and drive them with a single large motor.