Devnani–Eyyunni refined injectivity conjecture for elementary symmetric partitions

Let $\lambda$ be a partition, with length denoted by $\ell$, and let $\operatorname{pre}_j(\lambda)$ be the partition whose parts are all products of $j$ distinct parts of $\lambda$. Devnani–Eyyunni's refined injectivity conjecture. The function $\operatorname{pre}_j$ should be injective on partitions whenever $\ell>j$. The original injectivity claim fails when $\ell=j$, and this refined conjecture is also false: for every $j\geq3$, there are infinitely many pairs of distinct equal-size partitions of length $2j$ with the same $\operatorname{pre}_j$ image.