Super Pandigital Numbers

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Problem 571

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.

A n-super-pandigital number is a number that is simultaneously pandigital in all bases from 2 to n inclusively.
For example 978 = 11110100102 = 11000203 = 331024 = 124035 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?