Let $n$, $k$, $r_1, \dots, r_k$ be positive integers. For each $i \in [k]:=\{1,\dots,k\}$, suppose we are given $n$ permutations of the the set $[r_i]$, that is $f_1^{(i)}, \dots, f_n^{(i)}$ in $\mathfrak{S}_{r_i}$. Suppose that the group generated by these $n$ permutations acts transitively on $[r_i]$, i.e., it has a single orbit.
Now, for each $j \in [n]$, let $g_j := (f_j^{(1)}, \dots , f_j^{(k)}) \in \mathfrak{S}_{r_1} \times \dots \times \mathfrak{S}_{r_k}$, which acts on the finite ``box product'' set $B := [r_1] \times \dots \times [r_k]$ in the obvious way. Now consider the group $G$ of permutations of $B$ generated by $g_1, \dots, g_n$.
Question 1: Given the data $n$, $k$, $r_1, \dots, r_k$ and no information on the permutations $f_j^{(i)}$ (except the transitivity hypotheses above), what is the largest number of distinct orbits that the group $G$ may have?
The case $n=1$ is easy because then each $f_1^{(i)}$ must be a cyclic permutation of order $\mathrm{lcm}(r_1,\dots, r_k)$ and so there are $$ \frac{r_1 \dots r_k}{\mathrm{lcm}(r_1,\dots, r_k)} \tag{1} $$ distinct orbits.
For arbitrary $n$, note that $$ \frac{r_1 \dots r_k}{\max(r_1,\dots, r_k)} \tag{2} $$ is always an upper bound for the number of orbits, since each orbit projects in each $i$-th coordinate to the whole $[r_i]$.
Here is an example with $n=k=2$, $r_1=2$, $r_2=3$ where $f_1^{(1)} = \mathrm{id}$ and all the other permutations have order $2$. Then there is a single orbit:
This example shows that the upper bound (2) is not sharp. Incidentally, the bound (1) obtained for $n=1$ holds in this case. It's hard to imagine how more permutations (i.e., bigger $n$) could produce less orbits, so I ask:
Question 2: For $n \geq 2$, is the quantity (1) an upper bound for the number of orbits?
