Ah, from my experience using one of those for wooden tuning pegs, it won't do! The taper is not long enough, and so the tuning peg slips. The forces of metal strings on friction tuning pegs are even higher.
This is the kind of thing, around 30:1 taper:
TAPER PIN REAMER HSS 2MM 2.5MM 3MM 3.5MM...