Electrical canonical basis stratification conjecture for electroid varieties

Let $\mathcal{X}_n$ be the space of electrical networks, let $\mathcal{X}_\tau$ be the electroid variety indexed by a matching $\tau$ on $2n$ points, and let $\mathcal{X}_{\tau,>0}=\mathcal{X}_\tau\cap\mathcal{X}_{n,\geq 0}$. Let $E_{\mathbf{P}}$ be the proposed electrical canonical basis of $G_{d,n}$. Electrical canonical basis stratification conjecture. For every matching $\tau$, the elements $E_{\mathbf{P}}$ that do not restrict to zero on $\mathcal{X}_\tau$ form a basis of the homogeneous coordinate ring $\mathbb{C}[\mathcal{X}_\tau]$. Moreover, if $E_{\mathbf{P}}$ is not identically zero on $\mathcal{X}_\tau$, then it takes strictly positive values on $\mathcal{X}_{\tau,>0}$. This conjecture predicts that the electrical canonical basis behaves well under the electroid stratification, both algebraically through restriction and positively on the nonnegative strata.