A sumset version of a conjecture of Pilz

Abstract: Pilz's conjecture states that for any finite set $A={a_1,a_2,\dots,a_k}$ of positive integers and positive integer $n$ in the union of the sets ${a_1,2a_1,\dots,na_1},\dots, {a_k,2a_k,\dots,na_k}$ (considered as a multiset) at least $n$ values appear an odd number of times. In this short note we consider a variant of this problem. Namely, we show that in the sumset ${a_1,2a_1,\dots,na_1}+\dots+{a_k,2a_k,\dots,na_k}$ (considered as a multiset) at least $n$ values appear an odd number of times.