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It is known $AC^0[2]$ cannot get majority function.

Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$?

What is the best we found?

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Although not exactly what you're looking for, this paper seems to be the most current (2019) efforts towards tight $AC^0[2]$ bounds as of yet, and provides a good summary of the difference between $AC^0$ and $AC^0[2]$ bounds.

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