Unbounded balanced vertices for planar geodesic nets with four unbalanced vertices

Let a geodesic net in the Euclidean plane have four unbalanced vertices and some number of balanced vertices. Four-vertex unboundedness conjecture. There exist geodesic nets in the Euclidean plane with $4$ unbalanced vertices and an arbitrarily large number of balanced vertices. The assertion may already hold when the unbalanced vertices are the vertices of a square. This would show that no bound exists for the number of balanced vertices in terms of four fixed unbalanced vertices, extending the search beyond the known irreducible example with $16$ balanced vertices.