  Recent studies of pressure-driven flows of dilute polymer solutions in
straight channels demonstrated the existence of two-dimensional coherent
structures that are disconnected from the laminar state and appear through a
sub-critical bifurcation from infinity. These travelling-wave solutions were
suggested to organise the phase-space dynamics of purely elastic and
elasto-inertial chaotic channel flows. Here, we consider a wide range of
parameters, covering the purely-elastic and elasto-inertial cases, and
demonstrate that the two-dimensional travelling-wave solutions are unstable
when embedded in sufficiently wide three-dimensional domains. Our work
demonstrates that studies of purely elastic and elasto-inertial turbulence in
straight channels require three-dimensional simulations, and no reliable
conclusions can be drawn from studying strictly two-dimensional channel flows.
