Colbourn's conjecture on realizations with three largest equal subsquares

Let $(h_1h_2\dots h_k)$ be a partition with $h_1\geq h_2\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's conjecture. If $k\geq 3$, then an $\operatorname{RP}(h_1^3h_4\dots h_k)$ exists.

The conjecture concerns partitions whose three largest subsquares have the same size. The cases $k=3$ and $k=4$ follow from the known classification of realizations with at most four subsquares, and the cases $k\in\{5,6\}$ were proved by Colbourn; the general case remains open.