A sumset version of a conjecture of Pilz

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 $\{a_1,2a_1,\dots ,n{a}_1\},\dots ,\{a_k,2a_k,\dots ,n{a}_k\}$ (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 $\{a_1,2a_1,\dots ,n{a}_1\}+\cdots +\{a_k,2a_k,\dots ,n{a}_k\}$ (considered as a multiset) at least $n$ values appear an odd number of times.