Ballantine–Beck–Merca injectivity conjecture for elementary symmetric partitions

Let $n$ be a positive integer. A partition of $n$ is a weakly decreasing sequence of positive integers with size $n$ and length $\ell$. For $1\leq j\leq\ell$, define $\operatorname{pre}_j(\lambda)$ to be the partition whose parts are the products of all $j$-element subcollections of the parts of $\lambda$. Ballantine–Beck–Merca's injectivity conjecture. For $j\geq 2$, the function $\operatorname{pre}_j$ is injective on the set of partitions of size $n$ and length $\ell\geq j$. This conjecture is false: the $j=2$ case is solved, while the case $\ell=j$ has been disproved, and the present paper disproves the claim for $\ell=2j$ when $j\geq3$.