I didn't want to alter the cup bur in a way that made it not fit in my flex shaft when necessary.
I had a "make it work" moment and wrapped some tape around the end of the cup bur with a string. I made a loop around the end of the string to slip on a finger.
This actually worked really well. I was able to "drop" the cup bur in between wires while not losing it.
It was at the end of the string every time I needed it again! And I could easily remove the tape when I wanted to use the bur in my flex shaft.