Maximal $m \times 9n$ blocks of coloured beads, Part 1

Question

Let $n$ be a large positive integer. We consider $m \times 9n$ arrays of beads satisfying the following conditions:

1. Each bead is coloured one of three colours, say black, white, and green;
2. The number of black, white, and green beads in each row of the array is equal to $3n$;
3. For every $i < j$, colours $c = b,w,g$ and $1 \leq s \leq 9n$ put

$$\rho_{ij}^{(c)}(s) = \begin{cases} 1 & \text{if the beads at positions }is, js \text{ both have colour }c \\ 0 & \text{otherwise}. \end{cases}$$ and put $$\displaystyle \psi_{ij}^{(c)} = \sum_{s=1}^{9n} \rho_{ij}^{(c)}(s).$$ Then we require $$\displaystyle 3n \leq \Psi_{ij} = \psi_{ij}^{(b)} + \psi_{ij}^{(w)} + \psi_{ij}^{(g)} \leq 6n$$ for all $i < j$ and

4. For any pair of rows one cannot increase the value of $\Psi_{ij}$ by changing the colours of the beads from black to white, white to green, or black to green. Here we must recolour all of the beads of the same colour to another colour; in other words, we permute the colours. We are not allowed to recolour individual beads.

Call a $m \times 9n$ array acceptable if it satisfies the four conditions above. How does one estimate $m(n)$, the largest integer $m$ for which an acceptable array exists?

Note that $m(n)$ is well-defined, since the third condition guarantees that the rows of the array must be pairwise distinct, so the number of rows is certainly finite in any acceptable array.