$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$The braid groups $ B_1=1 $ and $ B_2\cong \mathbb{Z} $ have no perfect quotients.

$ B_3 $ has quotients $ \SL(2,\mathbb{Z}) $ and $ \PSL(2,\mathbb{Z})\cong B_3/Z(B_3) $. As a result, $ B_3 $ has infinitely many perfect quotients $ \SL(2,p) $ and $ \PSL(2,p) $ (of course $ p \neq 2,3 $ for it to be perfect).

Does this hold for all $ n\geq 3 $? In other words, do all braid groups $ B_n, n \geq 3 $ have infinitely many perfect quotients?

At the other extreme, are there any $ B_n $ other than $ n=1,2 $ that have no perfect quotients?