Electrical canonical basis conjecture

Let $G_{d,n}$ be the degree-$d$ homogeneous piece of the electrical network coordinate ring, let $\mathcal{C}^{(d)}_n$ index the proposed basis, let $E_{\mathbf{P}}$ denote its elements, let $L_P$ and $L_\sigma$ be grove coordinates, and let $\mathcal{X}_n$ be the space of electrical networks with nonnegative part $\mathcal{X}_{n,\geq 0}$. Let $\chi$ generate the cyclic group $\mathbb{Z}/2n\mathbb{Z}$ acting on $\mathcal{C}^{(d)}_n$ and $\mathcal{X}_n$. Electrical canonical basis conjecture. The space $G_{d,n}$ has an electrical canonical basis $E_{\mathbf{P}}$, $\mathbf{P}\in\mathcal{C}^{(d)}_n$, such that: for $d=1$, $E_P=L_P$; the basis elements take nonnegative values on the compactification of $\mathcal{X}_{n,\geq 0}$; every monomial in the grove coordinates $L_\sigma$ expands positively in the basis $E_P$; and the cyclic actions preserve $\mathcal{X}_{n,\geq 0}$ and satisfy

$$\chi^*(E_{\mathbf{P}})=E_{\chi(\mathbf{P})}.$$

This conjecture seeks an electrical analogue of Lusztig's dual canonical basis, with positivity and compatibility with cyclic symmetry; the source does not indicate which, if any, of these properties are known.