Gilbert–Pollak Steiner ratio conjecture

For a finite point configuration $P\subset\mathbb{R}^2$, let its Steiner ratio be the ratio of the cost of an optimal Steiner tree for $P$ to the cost of a minimum spanning tree for $P$. Gilbert–Pollak Steiner ratio conjecture. The minimum Steiner ratio over planar point configurations is

$$\sqrt{3}/2.$$

Gilbert and Pollak proposed this as the claim that the vertices of an equilateral triangle are the most efficient planar point configuration for Steiner trees. The statement is false: many regular simplices overlapping at a common vertex have a smaller Steiner ratio.