Matrix Coloring Dominant Extremal Cell Existence

The cells of a $(n^2 - n + 1) \times (n^2 - n + 1)$ matrix are coloured using $n$ colours. A colour is called *dominant* on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called *extremal* if its colour is dominant both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no extremal cells.