 Solution

Reflect the configuration in a plane perpendicular to the tube axis. The geometry and radial displacement are unchanged, whereas the imposed pressure gradient and every axial velocity reverse. By kinematic reversibility of Stokes flow, a radial migration velocity would also have to reverse; the spatial reflection leaves that component unchanged. Uniqueness of Stokes flow therefore forces it to vanish. Reflection in the meridional plane through the two axes similarly excludes azimuthal drift, so $c$ remains constant.