The homogeneous Landsberg–Berwald conjecture

A Landsberg space is a Finsler space whose Landsberg curvature vanishes, and a Berwald space is a Finsler space whose connection coefficients are independent of the direction. A Finsler space is homogeneous if its isometry group acts transitively on its points.

Homogeneous Landsberg–Berwald conjecture. Every homogeneous Landsberg space must be a Berwald space.

The conjecture is a special case of the unicorn problem, which asks whether every regular Landsberg space is Berwald. The paper proves this implication for Landsberg metrics of $(\alpha_1,\alpha_2)$-type, while the homogeneous case remains open in the stated context.