My son's 2012 Tiger is having similar problem. It will shift UP, but not DOWN, so hoping to find out the part(s) needed, not just replace the whole shift rod mechanism as some have said the dealer did, doesn't seem necessary.
I've already had it apart and put it back together. It was stuck in 6th but I removed the Detent Arm Spring and by hand easily turned the Gear Selector Drum down to neutral and first. I put it back together, sat on the bike and started it and shifted it back up through the gears to 6th, normal feel and shifting, but then would not shift down.
I suspect the Pivot Plate (photo of similar in post #4 of this thread). Is that what does the grunt work of turning the Detent Wheel for shifting? If this is the case would only one of the Pivot Plate's spikes be worn since it switches up okay?
The posts on the Pivot Plate looked okay from behind, but looking at the wear in post #4 I wouldn't have seen the wear area from the back, since I didn't remove the plate, just looked at it from the sides. Didn't have my C-Clamp removal tool with me.
I went in suspecting the pin was bent or broken off as has happened with many others, but it looked good. This bike is outside the VIN range of the spring replacement. My 2011 was in the range and had it done years ago.
Pivet Post #13 / Detent Wheel #5