Conditional local moves on polygonal networks for every n

Let $\Pi_n$ denote the polygonal resistor network used in the paper, and let a conditional local move be a local transformation of a subgraph of $\Pi_n$. Boundary edges and boundary spikes refer to the corresponding outer-boundary features; the outermost layer and outermost partial layer are the layers specified by the construction of $\Pi_n$.

Conditional local-move existence conjecture. For every integer $n\geq 3$, there exists a conditional local move on $\Pi_n$ such that:

$$\begin{cases} n\equiv 3\pmod 4: & \text{a boundary edge from the outermost layer is changed into a boundary spike on the opposite side};\\ n\equiv 0\pmod 4: & \text{a boundary spike from the outermost partial layer is changed into a boundary spike on the opposite side};\\ n\equiv 1\pmod 4: & \text{a boundary spike from the outermost partial layer is changed into a boundary edge on the opposite side};\\ n\equiv 2\pmod 4: & \text{a boundary edge from the outermost partial layer is changed into a boundary edge on the opposite side}. \end{cases}$$

The conjecture extends the conditional local moves listed earlier and suggests a combinatorial interpretation of their formulas in terms of groves or a grove-like structure. The source gives no proof of existence for all $n$, so the conjecture remains open.