I didn't want to alter the cup bur in a way that would prevent it from fitting in my flex shaft when needed.
So, I had a "make it work" moment.
I wrapped some tape around the end of the cup bur and attached a string.
Then I made a loop at the end of the string to slip over my 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.