On a conjecture of polynomials with prescribed range

We show that, for any integer m with 39 there exists a multiset M satisfying that 0∈M has the highest multiplicity q−m and ∑b∈Mb=0 such that every polynomial over the finite field Fq with the prescribed range M has degree greater than q−m. This implies that Conjecture 5.1 in Gács et al. (2010) [6] is false over any finite field Fq for p>9 and k:=m−1⩾3.