Your calculation of (5/9)(5/9)=25/81 (31%) is incorrect, as it assumes that the probability of beating each opponent is independent. On the contrary, these two events are quite correlated. I've calculated the probabilities for several of the 35 triples, and the probability of winning for the "best" die has ranged from 1/3 (33%) to 13/27 (48%). I doubt that any of the 35 cases yields less than 1/3.
Best wishes,Vadim Ponomarenko