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?