The one-local-maximum-per-way conjecture for polygon fillings

Let $G$ be the polygonal domain under consideration, and let $M(G)$ be its medial axis divided into $K$ pieces. A way $W$ is a distribution of $N$ discs over these pieces, written

$$W=\{n_i\}_1^{K},\qquad N=\sum_i^K n_i,$$

with $n_i\in\mathbb{N}$ and $n_i\in\{0,1\}$ when the $i$-th piece is a junction point.

One-local-maximum-per-way conjecture. There is at most one local maximum per way.

If true, an optimal $N$-disc filling could be found by generating a maximum for every way; this would reduce the search to a finite collection whose size is of order $O\left(N^{K-J-1}\right)$ when $J$ of the $K$ pieces are junctions. Whether a way can support multiple local maxima is not resolved in the supplied text.