  Recently, it has been argued by Kuklov et al., that unusual features
associated with the superflow-through-solid effect observed in solid He4 can be
explained by unique properties of dilute distribution of superfluid edge
dislocations. We demonstrate that stability of supercurrents controlled by
quantum phase slips (instantons), and other exotic infrared properties of the
superfluid dislocations readily follow from a one-dimensional quantum liquid
distinguished by an effectively infinite compressibility (in the absence of
Peierls potential) associated with the edge dislocation's ability to climb.
This establishes a new class of quasi-one-dimensional superfluid states that
remain stable and long-range ordered despite their low dimensionality. We
propose an experiment to test our mass-current--pressure characteristic
prediction.
