  Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph
$G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners,
states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$
vertices with chromatic number $k\geq 4$. In this paper, we study the same
problem with the additional constraint that $G$ is $\ell$-connected. For
$2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le
(k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved
if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an
asymptotically tight upper bound \[ P_G(k) \le k!(k-1)^{n-\ell - k + 1} +
O((k-2)^n), \] and provide a matching lower bound construction. For the ranges
$k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph
maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to
connected graphs with minimum degree $\delta$.
