Determine the minimum size of a non-winnable lattice

Determine the size of the smallest finite lattice on which the toggle game cannot be won.

Background

The paper proves non-constructively that a finite lattice exists on which the toggle game cannot be won, thereby refuting the left-linear strengthening of the Non-Cancelling Intersections conjecture. Its construction proceeds through a probabilistic marking argument over an affine plane and yields only an extremely large upper bound on the size of a counterexample.

The authors emphasize that their estimate is an upper bound for the particular proof construction, not a lower bound for all counterexamples. The unresolved problem is therefore to determine, or substantially bound, the minimum number of vertices of any finite lattice with an unwinnable toggle game.