Counterexamples regarding elementary symmetric partitions

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre${}_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre${}_j$ is injective on the set of partitions of $n$ with length $\mathrm{\ell }\ge j$. The $\mathrm{\ell }=j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\mathrm{\ell }>j$. In this article, we answer this refined conjecture in the negative by proving that pre${}_j$ is not injective on partitions of $n$ with length $2j$ for $j\ge 3$. We also prove that the analogous map prh${}_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.