so you do not have a nose at all? My opinion and thinking only, if the stringer was not designed for the actual nose to fasten to a lip of the stringer and to allow optimal support, a after market nose will not hold. Nose's are the one part of a stair way that takes all the weight of the person going up and down. Without proper fastening of a proper nose, it will not stay.
We must be on the same page with the defination of a NOSE.
Second, the quarter round sounds like the best choice and it can wrap all the way around as long as its not part of the nose.