Nabutovsky's unbounded balanced-vertex conjecture for geodesic nets

A geodesic net is a finite embedded graph in the Euclidean plane whose edges are geodesic segments; its vertices are called unbalanced or balanced according to whether the incident edge directions fail or satisfy the balancing condition. The boundary vertices are the unbalanced vertices.

Nabutovsky's conjecture. There exist geodesic nets in the Euclidean plane with $4$ unbalanced vertices and an arbitrarily large number of balanced vertices. Moreover, this may already hold when the unbalanced vertices are the vertices of a square.

This conjecture asks whether the number of balanced vertices can be unbounded when there are four boundary vertices. The paper constructs an irreducible example with $25$ balanced vertices, improving the previous record of $16$, but does not establish arbitrarily large numbers.