Pandigital square numbers

A square number containing every digit from 0 to 9 exactly once.
Chalkdust Prize Crossnumber #1, 5 down

Whilst trying to answer 5 down in the Chalkdust crossnumber, I discovered that there are 87 square numbers in base 10 that use every digit exactly once (without a leading zero). This made me wonder how many square numbers have this property in other number bases.

I quickly wrote some code to get an idea of how many pandigital (without repeating digits or a leading zero) squares there are in other number bases.

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