This post derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the bounding constant (in the post, it is $x_n$) for the case where the binomial r.v.'s may not be identically distributed?

Mathematically, let $X_i\stackrel{indep}{\sim} Bin(n,p_i)$, $1\leq i\leq m$.

Is it possible to find a $c_{m,n}$, such that $P(\max_{1\leq i\leq m} X_i > c_{m,n})\to 0$ as $m,n\to\infty$, also assuming that $m=\mathcal{O}(n^r)$ for some $r>1$?

The condition at the end lets $m$ get larger at a rate faster than $n$. My gut also tells me that $c_{m,n}$ will depend on some function of the $p_i$'s (in addition to $m$ and $n$).