Colbourn's conjecture on realizations under a largest-part bound

Let $(h_1\dots h_k)$ be a partition with $h_1\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's conjecture. If

$$k\geq 5\quad\text{and}\quad 0<h_1\leq (k-2)h_k,$$

then an $\operatorname{RP}(h_1\dots h_k)$ exists.

This is the second unconditional-existence family conjectured by Colbourn. It extends the known results on realizations with few subsquares or with at most two distinct part sizes; no resolution of the full stated range is given here.