  We present a construction of two infinite graphs $G_1$ and $G_2$, and of an
infinite set $\mathscr{F}$ of graphs such that $\mathscr{F}$ is an antichain
with respect to the immersion relation and, for each graph $G$ in
$\mathscr{F}$, both $G_1$ and $G_2$ are subgraphs of $G$, but no graph properly
immersed in $G$ admits an immersion of $G_1$ and of $G_2$. This shows that the
class of infinite graphs ordered by the immersion relation does not have the
finite intertwine property.
