3 cables should work. Each cable length is a single parameter that defines a sphere around the mount, and the intersection of three spheres defines a point.

I suspect the reason they chose 4 is that 3 cables would limit you to a triangular region, so you’d only be able to access half the room.

You can join two of the cables together that come from each corner unit into one. each unit - still with an eye hole in another corner - moves the cable left or right. With the opposite corner unit, you can pick X,Y on the ceiling. I'd put the third cable in the grabber, which can be used to lower it to the floor.

Probably if you went this way, you'd want to measure each ceiling cable so it is about X+Y length, or otherwise the slack needs to be held somewhere.

This is one less actuator than their system if I understand them, and they are not counting the gripper itself.

Range of motion would be confined to the space beneath the triangle bounded by the three anchor points. Most spaces are rectangular, so bounding it to four corners would seem likely to give a more useful range of motion.

Me, i'm wondering if the coreXY kinematics could be adapted to make do with only two actuators on four anchors.