Ballantine, Beck, and Merca defined a map $\mathrm{pre}_2$, which sends an integer partition $λ= (λ_1, \dots, λ_{\ell})$ to the set $\{λ_iλ_j : 1 \leq i < j \leq \ell\}$ consisting of the pairwise products of parts of $λ$. The same three authors and Sagan conjectured that for each $n$, the map $\mathrm{pre}_2$ is injective on the set of integer partitions of $n$. In this note, we prove their conjecture.