The Extended 1-2-3 Conjecture of Pilz

Let $A$ be a finite subset of the positive integers, and let $n$ be a positive integer. For each positive integer $k$, write $k\boldsymbol{\bullet} A=\{ka:a\in A\}$, and let $\Delta$ denote symmetric difference of sets. Extended 1-2-3 Conjecture. One has

$$\left|A\Delta (2\boldsymbol{\bullet} A)\Delta\cdots\Delta (n\boldsymbol{\bullet} A)\right|\geqslant n.$$

Pilz proved the inequality for $n\leq 6$, and Huang, Ke and Pilz later formulated it as the Extended 1-2-3 Conjecture. The source paper states that it resolves the conjecture for all sufficiently large $n$, so the assertion is not open in that range; the supplied text does not specify the exact threshold or whether all remaining cases have been settled.