Modified Ballantine–Beck–Merca injectivity conjecture for $pre_k$

Let $\mathcal{P}(n)$ denote the set of integer partitions of $n$. For a partition $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$, let $pre_k(\lambda)$ be the partition whose parts are the products $\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{i_k}$ over all $1\leq i_1<i_2<\cdots<i_k\leq\ell$.

Modified Ballantine–Beck–Merca conjecture. For $k\geq 3$, $pre_k$ is injective on partitions of $n$ with $j$ parts for each $j\geq k+1$.

The modification excludes the $k$-part case, where the original conjecture is false. The source gives the modified statement after observing that equal images must have the same number of parts, since $\binom{\ell}{k}$ is strictly increasing in $\ell$ for fixed $k$; its resolution is not supplied here.