Let $G_1, G_2 $ and $G_3$ be subgroups of $G$ with finite indexes. Suppose that there are $x_1,x_2,x_3\in G$ such that

- $G_1x_1\cap G_2x_2=G_2x_2\cap G_3x_3=G_1x_1\cap G_3x_3=\emptyset$,
- $G_1G_2x_2\cup G_1G_3x_3=G_2x_2\cup G_3x_3$,
- $G_2G_1x_1\cup G_2G_3x_3=G_1x_1\cup G_3x_3$, and
- $G_3G_1x_1\cup G_3G_2x_2=G_1x_1\cup G_2x_2$.

Must it be that at least two of $|G:G_1|$, $|G:G_2|$, $|G:G_3|$ are equal?