  \noindent Azarpendar and Jafari in 2020 proved the following inequality
$$\chi \left( {\rm KG} ^r ({\cal F} , s) \right) \geq \left\lceil \frac{ {\rm
ecd}^r \left( {\cal F} , \left\lfloor \frac{s}{2} \right\rfloor \right) }{r-1}
\right\rceil ,$$ and noted that it is plausible that the above inequality
remains true if one replaces $\left\lfloor \frac{s}{2} \right\rfloor$ with $s$.
  \noindent In this paper, considering the relation ${\rm ecd}^r \left( {\cal
F} , x \right) \geq {\rm cd}^r \left( {\cal F} , x \right)$ which always holds,
we show that even in the weaker inequality $$\chi \left( {\rm KG} ^r ({\cal F}
, s) \right) \geq \left\lceil \frac{ {\rm cd}^r \left( {\cal F} , \left\lfloor
\frac{s}{2} \right\rfloor \right) }{r-1} \right\rceil ,$$ no number $x$ greater
than $\left\lfloor \frac{s}{2} \right\rfloor$ could be replaced by
$\left\lfloor \frac{s}{2} \right\rfloor$.
