Subpower membership conjecture for algebras with few subpowers

Let $\mathbb{A}$ be a finite algebra, and say that it has few subpowers when the equivalent conditions in the source's Few Subpowers theorem hold. Given a finite subset $S\subseteq \mathbb{A}^n$ and an element $x\in \mathbb{A}^n$, the subpower membership problem asks whether $x$ belongs to the subalgebra of $\mathbb{A}^n$ generated by $S$. Few-subpowers subpower-membership conjecture. If $\mathbb{A}$ has few subpowers, then the subpower membership problem can be solved in polynomial time. The source presents this as an open question; general instances can be EXPTIME-complete, while stronger hardness results cover many algebras without few subpowers.