Holomorphicity conjecture for center distributions on Kähler manifolds

Let $M$ be a Kähler manifold and let $f$ be a holomorphic partially hyperbolic diffeomorphism of $M$, with center distribution $E^c$. Kähler center-holomorphicity conjecture. The center distribution $E^c$ is holomorphic. This is a stronger regularity expectation in the Kähler setting, motivated by the paper's examples, most of which are not on Kähler manifolds; the source does not state a resolution.