I've just made a new test chamber, The Chessboard Revised, where the problem is fixed. I just raised the screen shields higher, and that solved the problem.

Try putting one or two of the caged cubes in Glass boxes. That way the laser can still get in, but the player can't move them around.

Thanks for the input, I'm working on a way to block it off. I'm thinking about light beams that are turned off when the cubes are dropped.

This test was REALLY fun. the only down side is, the player can get in and re direct the "caged cubes" To pint right where it needs to go, and avoid all your set up, sadly. But it was still fun as hell to play though!