Eccoci.

A positive number is pandigital in base $b$ if it contains all digits from $0$ to $b - 1$ at least once when written in base $b$.

An $n$-super-pandigital number is a number that is simultaneously pandigital in all bases from $2$ to $n$ inclusively.
For example $978 = 1111010010_2 = 1100020_3 = 33102_4 = 12403_5$ is the smallest $5$-super-pandigital number.
Similarly, $1093265784$ is the smallest $10$-super-pandigital number.
The sum of the $10$ smallest $10$-super-pandigital numbers is $20319792309$.

What is the sum of the $10$ smallest $12$-super-pandigital numbers?