Conjecture on maximizers of the partition discrepancy

For partitions $\beta$ and $u$, let $\beta\sqcup\nu$ denote the partition obtained by distributing the parts of $\beta$ and $u$ among the parts of a partition $\bar\lambda$ of $n$, and let $\Delta(\lambda,\mu,\nu)$ be the quantity defined in the source. The maximum of $\Delta(\lambda,\mu,\nu)$ over triples with $\lambda\vdash n$, $\mu\vdash n-k$, and $\nu\vdash k$ is attained at a triple of partitions satisfying the following condition.

Maximizer conjecture. At most one part of $\lambda$ is the sum of a part in $\mu$ and a part in $\nu$, while the other parts of $\lambda$ are distributed in $\mu$ and $\nu$.

This conjecture proposes a structural description of a maximizer for the discrepancy $\Delta$; the supplied text gives no resolution, so its status remains open.