Wardrobe Under Shelf LED lighting that can be moved about freely
Just an idea - Why not power every vertical wall mounted bracket with DC - the first bracket positive, second ground, then positive, ground etc.
Each shelf rail connecting to the bracket carries the current along to the edge.
(Either via cable or possibly through the metal of the rail itself for the ground connections)
Use a rectifier then at the shelf edges to correctly orientate the power for the led strip.
Much easier then as there is just the one contact to be made at each joint....