  The main result of this paper shows that if $\mathcal{M}$ is a consistent
strong linear Maltsev condition which does not imply the existence of a cube
term, then for any finite algebra $\mathbb{A}$ there exists a new finite
algebra $\mathbb{A}_\mathcal{M}$ which satisfies the Maltsev condition
$\mathcal{M}$, and whose subpower membership problem is at least as hard as the
subpower membership problem for $\mathbb{A}$. 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 ($k \geq 3$) whose subpower
membership problem is EXPTIME-complete.
