Equivalence of the new intermediate condition and the Kähler–Finsler condition

Determine whether the condition G_k(z;v)(Γ^k_{j;i}(z;v)−Γ^k_{i;j}(z;v))=0 for all nonzero tangent vectors and all indices coincides with the Kähler–Finsler condition for strongly pseudoconvex complex Finsler metrics.

Background

The paper introduces an intermediate condition between the strongly Kähler–Finsler and weakly Kähler–Finsler conditions by removing the contraction with v^j from the weakly Kähler–Finsler condition. The authors prove that this new condition is equivalent to the weakly Kähler–Finsler condition, including in complex dimensions greater than two.

The remaining issue is whether this intermediate condition is also equivalent to the Kähler–Finsler condition. The paper does not resolve that general equivalence; instead, it later establishes examples showing that weakly Kähler–Finsler and Kähler–Finsler metrics can differ.