Fewest d6 and d8 so neither total is favoured

Alice rolls $m$ fair six-sided dice and Bob rolls $n$ fair eight-sided dice; each adds up their own faces. Find the smallest positive integers $m^*$ and $n^*$ for which Alice's total is exactly as likely to be below Bob's total as above it, and report the product $m^*n^*$.

$63$. Both totals have distributions symmetric about their means $3.5m$ and $4.5n$, so the difference $S-T$ is symmetric about $3.5m-4.5n$ and the two probabilities match when that is $0$. Then $7m=9n$, so the smallest choice is $m=9$, $n=7$.