One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by [Venkataramana that Burau representations][1] of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument.

This should follow from the approach to prove Theorem 1.2 of [Masbaum-Reid.][2] (Alan Reid suggested this to me)



  [1]: https://doi.org/10.4007/annals.2014.179.3.4
  [2]: https://msp.org/gt/2012/16-3/p04.xhtml