Conjecture on maximizers of r-star for n divisible by four

Let $n$ be a positive integer and let $r_\star$ be the maximum value of the $r$-loading over triples of partitions of $n$. For a partition, notation such as $(k^4)$ means four parts equal to $k$, and $(2^{2k})$ means $2k$ parts equal to $2$.

Maximizer conjecture for $r_\star$. When $n=4k$ with $k\ge 2$, the values $r_\star$ are attained by

$$\mathbf t=((k^4),(2^{2k}),(2^{2k})).$$

The conjecture is inferred from the cases $n=8,12,16,20$ displayed in the paper; no proof or resolution is supplied.