Conjecture on maximizers of b-star

Let $n$ be a positive integer and let $b_\star$ be the maximum value of the $b$-loading over triples $\mathbf t=(\lambda,\mu,\nu)$ of partitions of $n$.

Maximizer conjecture for $b_\star$. For $n\ge 6$, the values $b_\star$ are attained by triples such that

$$\lambda=\mu\quad\text{or}\quad\mu=\nu.$$

Moreover, when $n=3k$ with $k\ge 2$, the values $b_\star$ are attained by triples such that

$$\lambda=\mu=\nu.$$

The conjecture is based on the tabulated cases, particularly $n=6,9,12,15,18$; no proof or resolution is supplied.