I had similar behavior with my passenger window. The window would try to drop when the door handle was pulled, but wouldn’t move far enough down to clear the top. During all my research I haven’t come across any internet info for window regulators with built in adjustment for cable tension.
I tried the cable tie fix and it wasn’t stable enough. The green housing end was popping out of its receptacle. I tired adding an inline bicycle brake adjuster, but I couldn’t get enough slack to get it in place. During all messing around I ended up dislodging the cable from a pulley. Then I cracked the pulley trying to get it back on. After that the only option was a new window regulator.
The cable tension on mine was snug to tight. It shouldn’t matter if the window is up or down. The cable tension should stay fairly constant due to the design.
The green piece may be one of the cable housing ends. Probably from upper portion where it’s hard to see.
