Can $p_n\#<p_{n+1}^{p_{n+1}}$ be extended to $c_n<p_{n+1}^{p_{n+1}}$ for colossally abundant $c_n$?

Question

Trivially, we have $n!\le n^n$. So, for prime $p$, we have $p!\le p^p$. From here it is easy to see that for the $n$th primorial and the $(n+1)$th prime,

$p_n\#<p_{n+1}^{p_{n+1}}$

My question is: Can this result be extended to $n$th colossally abundant number, $c_n$, such that

$c_n<p_{n+1}^{p_{n+1}}$

??