Engbers–Erey–Fox–He conjecture for chromatic polynomials of l-connected graphs

Let $G$ be a $k$-chromatic $l$-connected graph on $n$ vertices, where $k\geq 4$ and $l\geq 3$, and let $P(G,x)$ denote its chromatic polynomial. Engbers–Erey–Fox–He conjecture. For every $x\geq k$,

$$P(G,x)\leq (x)_k(x-1)^{n-l-k+1}+O\big((x-2)^n\big).$$

This is a proposed extension of the preceding chromatic-polynomial bounds from $2$-connected to $l$-connected graphs. The source says that the case $x=k$ was proved, while the displayed estimate for all $x\geq k$ is presented as a conjectural generalization.