The possibility of using Shattered Pellet Injection(s) after the Thermal Quench phase of an ITER disruption in order to deplete Runaway Electron (RE) seeds before they can substantially avalanche is studied. Analytical and numerical estimates of the required injection rate for shards to penetrate into the forming RE beam and stop REs are given. How much material could be assimilated before the Current Quench (CQ) becomes too short is also estimated. It appears that, if Hydrogen pellets were used, the required number of pellets to be injected during the CQ would be prohibitive, at least considering the present design of the ITER Disruption Mitigation System (DMS). For Neon or Argon, the required number of pellets, although large, might be within reach of the ITER DMS, but the assimilated fraction would have to be very small. Other materials may be better suited but would require a modification of the ITER DMS.