The eventual junction-occupation conjecture for all local-maximal polygon fillings

Let $G$ be the polygonal domain under consideration and let $M(G)$ be its medial axis, including its junction points. A filling solution that is a local maximum is a filling by $N$ discs whose filling measure is locally maximal.

Eventual all-local-maxima junction-occupation conjecture. For a given $G$ and $M(G)$, there is an $N'$ such that for $N\geq N'$, the junction points are always occupied in all filling solutions that are local maxima.

This is explicitly proposed as a stronger version of the conjecture for optimal filling solutions, and would ensure that every sufficiently large local maximum occupies all junction points. The supplied text gives no proof or resolution status.