The main result of this paper shows that if M is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra A there exists a new finite algebra AM which satisfies the Maltsev condition M, and whose subpower membership problem is at least as hard as the subpower membership problem for A. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence k-permutable (k ≥ 3) whose subpower membership problem is EXPTIME-complete.2010 Mathematics Subject Classification. 08B05, 68Q17.