Injectivity of pairwise products of partition parts

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_2(\lambda)$ be the partition whose parts are the products $\lambda_i\lambda_k$ for $1\leq i<k\leq\ell$. Pairwise-product injectivity conjecture. If $\lambda$ and $\mu$ are partitions of $n$ for some $n\geq0$, then

$$\operatorname{pre}_2(\lambda)=\operatorname{pre}_2(\mu)\quad\Longleftrightarrow\quad\lambda=\mu.$$

This injectivity would imply the first odd-part dominance inequality above. It is presented as conjectural, with no resolution given in the source.