The cube-term conjecture for the Hidden Kernel Problem

Let $\mathbb{A}$ be a finite algebra, and let a cube term be a term operation $t(x_1,\dots,x_n)$ such that, for every $i\in[n]$, there is a choice of $u_1,\dots,u_n\in\{x,y\}$ with $u_i=y$ for which $t(u_1,\dots,u_n)=x$ holds in $\mathbb{A}$. Write $\operatorname{HKP}(\mathbb{A}^n)$ for the Hidden Kernel Problem over the $n$-th power of $\mathbb{A}$. Cube-term conjecture. If $\mathbb{A}$ has a cube term, then $\operatorname{HKP}(\mathbb{A}^n)$ has an efficient quantum solution. The conjecture proposes that the regularity supplied by a cube term is sufficient for efficient quantum algorithms, whereas counting arguments cannot rule out such algorithms because subalgebras, and hence congruences, of powers of $\mathbb{A}$ have polynomially bounded generating sets. An efficient quantum solution would in particular give an efficient quantum solution to the hidden normal subgroup problem.