Generalizing Efron, we can get probability $\ge 70\%$ with six 10-sided dice:

 - 10 sides $=6$
 - 3 sides $=11$, 7 sides $=5$
 - 4 sides $=10$, 6 sides $=4$
 - 5 sides $=9$, 5 sides $=3$
 - 6 sides $=8$, 4 sides $=2$
 - 7 sides $=7$, 3 sides $=1$

We can get probability $\ge 72\%$ (actually $.7218934911 = 122/169$) with eight 130-sided dice:

 - 130 sides $=8$
 - 36 sides $=15$, 94 sides $=7$
 - 50 sides $=14$, 80 sides $=6$
 - 58 sides $=13$, 72 sides $=5$
 - 65 sides $=12$, 65 sides $=4$
 - 72 sides $=11$, 58 sides $=3$
 - 80 sides $=10$, 50 sides $=2$
 - 94 sides $=9$, 36 sides $=1$