We imagine the binary pair moving through a field of distant trans Neptunian objects. Long lived weak resonances with swarms of smaller bodies can gently perturb the binary orbits. Over very long times these perturbations act as a stirring spoon that maintains a small orbital eccentricity.
In effect the outer system supplies occasional nudges that stop the binary from fully circularising. This keeps the tidal heating engine running even as internal dissipation tends to damp the orbits.