Kenyon–Wilson's response-matrix characterization conjecture for lower-dimensional electrical-network cells

Let $EP_n$ be the poset of cells of circular electrical networks, and let $M\in\mathrm{Mat}_{n\times n}(\mathbb{R})$. A cell has co-dimension $r$ when its codimension in $EP_n$ is $r$. For sets $S_1$ and $S_2$ of circular minors, write $|S_1|=\frac{n(n-1)}{2}-r$ and $|S_2|=r$. The first two conditions from Theorem

are the conditions stated there for a response matrix. **Kenyon–Wilson's conjecture.** The matrix $M$ is the response matrix of a circular electrical network belonging to a cell of co-dimension $r$ in the poset $EP_n$ if and only if the first two conditions from Theorem

are satisfied and there exist sets $(S_1,S_2)$ such that the circular positivity of all circular minors from $S_1$ and the vanishing of all circular minors from $S_2$ imply circular non-negativity of all other circular minors.

This conjecture gives a proposed positivity-and-vanishing test for identifying the lower-dimensional cells of the electrical-network cell poset. The supplied text does not state whether it has been resolved, so its status is left open.