  In the experiment on superfluid transport in solid He4 [PRL {\bf 100}, 235301
(2008)], Ray and Hallock observed an {\it anomalously large isochoric
compressibility}: the supersolid samples demonstrated a significant and
apparently spatially uniform response of density and pressure to chemical
potential, applied locally through Vycor "electrodes". We propose that the
effect is due to {\it superclimb} : edge dislocations can climb because of mass
transport along superfluid cores. We corroborate the scenario by {\it ab
initio} simulations of an edge dislocation in solid He4 at $T=0.5K$. We argue
that at low temperature the effect must be suppressed due to a crossover to the
smooth dislocation.
