The nearby-optimal-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. For $N-1$ discs, write the optimal way as $\{n'_i\}_1^{K}$, and for $N$ discs write a way as $\{n_i\}_1^{K}$. The distance between pieces is counted by the number of connecting pieces.

Nearby-optimal-way conjecture. Given the optimal way of distributing $N-1$ discs, $\{n'_i\}_1^{K}$, the optimal way of distributing $N$ discs is nearby, where nearby means

$$\sum_1^K \mid n_i-n'_i\mid$$

is small; moreover, if the number of discs assigned to a piece decreases, then the pieces whose disc counts increase have minimal distance to the decreased piece.

This conjecture would justify using the optimal $(N-1)$-disc solution to restrict the search for the optimal $N$-disc solution to a small neighborhood of ways. The supplied text gives no resolution status.