Tomescu's conjectural coloring bound for ℓ-connected graphs

Let $G$ be a $k$-chromatic, $\ell$-connected graph on $n$ vertices, with $k\geq 4$ and $\ell\geq 3$. Let $x\geq k$ be an integer, let $P_G(x)$ be the number of proper $x$-colorings, and let $(x)_k=x(x-1)\cdots(x-k+1)$.

Tomescu's ℓ-connected generalization.

$$P_G(x)\leq (x)_k(x-1)^{n-\ell-k+1}+O((x-2)^n).$$

This generalizes the preceding conjectures to $\ell$-connected graphs. The source notes that its theorem proves the case $x=k$, but does not report a resolution for general integer $x\geq k$.