In the experiment on superfluid transport in solid 4He [Phys. Rev. Lett. 100, 235301 (2008)], Ray and Hallock observed an 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 superclimb: edge dislocations can climb because of mass transport along superfluid cores. We corroborate the scenario by ab initio simulations of an edge dislocation in solid 4He at T=0.5  K. We argue that at low temperature the effect must be suppressed due to a crossover to the smooth dislocation.