On a conjecture of polynomials with prescribed range

We show that, for any integer $\mathrm{\ell }$ with $q-\sqrt{p}-1\le \mathrm{\ell }<q-3$ where $q=p^n$ and $p>9$, there exists a multiset $M$ satisfying that $0\in M$ has the highest multiplicity $\mathrm{\ell }$ and ${\sum }_{b\in M}b=0$ such that every polynomial over finite fields $\text{fq}$ with the prescribed range $M$ has degree greater than $\mathrm{\ell }$. This implies that Conjecture 5.1. in \cite{gac} is false over finite field $\text{fq}$ for $p>9$ and $k:=q-\mathrm{\ell }-1\ge 3$.