Gao–Lam–Xu's electrical canonical basis conjecture

Let $G_{m,n}$ be the degree-$m$ part of the grove algebra, and let $c$ generate the cyclic action

$$\langle c\rangle\simeq\mathbb{Z}/2n\mathbb{Z}.$$

The basis elements are to be indexed by height $m$ staircase plane partitions of size $n-1$. Gao–Lam–Xu's electrical canonical basis conjecture. For all $m\geq 1$, $G_{m,n}$ has an electrical canonical basis which, among other properties, has those indices, and the induced action of $\langle c\rangle$ corresponds on the basis elements to promotion of staircase plane partitions. Such a basis would provide the representation-theoretic mechanism for cyclic sieving for promotion in all heights; its existence is conjectural.