I will try to help you with this question.
A thin line down the screen,all way,which does not change with the picture content or aspect,can be caused only by two causes.
#1 and the most common one would be a failure of the corresponding TCP ic,this is the drive IC that is inside of the ribbon cable for that specific area of the panel.
This is not a replaceable part,it is bonded with the plasma panel in the manufacturing process.
What that means ,is the plasma panel will need to be replaced!
This will not be economical outside of manufacturer warranty since the panel cost will be around-$1300-way exceeding the cost of a new unit.
#2 possible failure is one of the C boards(corresponding the affected area in the picture).This failure is a lot more rare than the plasma panel(TCP IC) and most of the time will affect a larger area, that will create a larger error on the screen,more like a strip ,than a line!
In case this corresponding C board is bad,it will cost about-$220-$250 to have it replaced.
Based on my extended experience with these Panasonic plasma TVs,I would say about 90% of the cases it turns out to be the plasma panel.
I hope this helps you getting an idea and making at least an informed decision about this set.Please feel free and ask me any additional related questions you may have,I will be happy to assist you.
Thank you and have a good day!