Found this link as my phone recently developed this pink/purple line issue. I have also noticed that they pretty much develop in the same basic location on every affected phone. Long story short, Samsung claims damage to the display, due to being dropped or whatever. However, many users claim that they never have dropped their phone. Both seem to be 'sort of' correct. It seems to be caused by something coming loose inside the phone, and can be exacerbated by a drop, but also happens simply due to the day-to-day usage of the phone and the inherent jostling it experiences.
The current accepted 'fix' of going in to the test screen and 'resetting' the red, green, and blue actually does nothing, which is why some people have to do it many times to see a result. Ok, so it actually doesnt do 'nothing', per se, but it is not the tests solving the problem beyond causing users to press a specific area of the screen. You can achieve the same results no matter what screen is displayed at the time, although something that prevents an action/menu from bring triggered by the press is best. I usually just use the 'recent apps' screen. You can press slightly below the selfie camera, on the actual screen, and when touching the correct area you will see the line disappear. It seems to only take a light touch, so dont press TOO hard and crack your screen. You may have to do it multiple times (sometimes a LOT), but it should provide at least temporary relief without needing to send your phone in or take it to a repair center. This will also hopefully help the users who can not get the test screen to display.
Hopefully Samsung will come up with a better response than they currently have thus far...