  In 2015, Brown and Erey conjectured that every $2$-connected graph $G$ on $n$
vertices with chromatic number $k\geq 4$ has at most
$(x-1)_{k-1}\big((x-1)^{n-k+1}+(-1)^{n-k}\big)$ proper $x$-colorings for all
$x\geq k$. Engbers, Erey, Fox, and He proved this conjecture for $x=k$. In this
paper, we prove Brown and Erey's conjecture under the condition that either the
clique number of $G$ is $k$, or the independent number of $G$ is $2$.
