  We argue that the three key phenomena recently observed in solid
$^4$He---mass supertransport, anomalous isochoric compressibility (syringe
effect), and giant plasticity---are closely linked to each other through the
physics of an interconnected network of tilted quantum-rough dislocations. As
immediate implications of this connection several predictions follow: In the
absence of $^3$He impurities, the syringe effect and giant plasticity persist
down to $T=0$; the dynamical low-frequency syringe and giant-plasticity
responses are dispersionless; and similarly to giant plasticity but without
direct relationship to the supertransport along the dislocation cores, $^3$He
impurities should suppress the syringe effect partially or completely at
appropriately low temperatures.
