Nonexistence of a finite planar fair-win goal set

In the planar achievement game, a finite point set $G\subset\mathbb{R}^2$ is used as the goal set: players alternately place points, and Player 1 seeks to build a congruent copy of $G$ before Player 2 does. A fair win is a win in which Player 1 builds a goal set on a turn preceding every turn on which Player 2 builds one. Planar fair-win nonexistence conjecture. There is no finite point set $G\subset\mathbb{R}^2$ for which Player 1 has a strategy ensuring a fair win in the $G$-achievement game played in the plane. The preceding examples show that fair wins can occur for finite hypergraphs, but the claim asserts that no finite planar point set has this property.