Hacking–Kim conjecture on nonsymmetric Kähler–Einstein Fano polygons

Let $P$ be a Fano polygon, that is, a two-dimensional Fano polytope. Call $P$ symmetric when its symmetry group is nontrivial. Hacking–Kim conjecture. If $P$ is a Kähler–Einstein Fano polygon and is not symmetric, then $P$ is a triangle.

Every known Kähler–Einstein Fano polygon that is not symmetric is a triangle, while the theorem established in the paper proves the corresponding barycentric-transformation statements for Kähler–Einstein polygons and symmetric polygons. The conjecture concerns whether all nonsymmetric Kähler–Einstein Fano polygons arise as triangles.