I used a computer program to search for voting profiles that have the desired dominance properties (every group of three candidates has a fourth that beats all of them more than half the time).
I found one with 23 candidates and 9 voters.
(None were found with 19 or 21 candidates with the symmetry properties that I used to reduce the scope of the search.)
This yields a set of 23 dice with 9 faces where every set of three dice has a fourth that beats all three, on average.
But how do we handle 9-sided dice?