Existence of non-ball strongly convex domains with Kähler–Finsler Kobayashi metric

Construct or determine whether there exists a smooth bounded strongly convex domain that is not biholomorphically equivalent to the unit ball and whose Kobayashi metric is Kähler–Finsler.

Background

The paper proves that the Kobayashi metric of the smooth bounded strongly convex domain Ω={z∈C²: |z₁|²+|z₂|²+t|z₁|⁴<1}, with t>0, is weakly Kähler–Finsler but not Kähler–Finsler. This gives a smooth non-Kähler–Finsler example and provides a negative answer for that particular domain.

The authors then reduce the broader existence problem to whether some smooth bounded strongly convex domain other than the unit ball can nevertheless have a Kähler–Finsler Kobayashi metric. A positive answer would produce the first non-Kähler–Berwald, Kähler–Finsler example.