Yes, it is on its own circuit. You don't think the proximity is a problem do you?
If it were interference from the refrigerator, it would appear whenever the compressor is running, and disappear when it goes off. Since your problem seems to be more or less constant, I would not suspect the reefer.
Sounds like a bad monitor. The LCD's have a grid where the vertical and horizontal lines are controlled by elements along the screen edges. One bad element on the top or bottom will produce a vertical line.