Conjecture on k-positive symmetrized curvature operators and projective space

Let $(M^n,\omega)$ be a compact Kähler manifold. The symmetrized curvature operator $\mathcal R$ is $k$-positive if its relevant curvature positivity condition holds for $k$ directions.

Symmetrized curvature conjecture. If $1\leq k\leq \left[\frac{n}{2}\right]$ and $M$ has $k$-positive symmetrized curvature operator $\mathcal R$, then $M$ is biholomorphic to $\mathbb{C}\mathbb{P}^n$.

This proposes a characterization of complex projective space through positivity of the symmetrized curvature operator. The supplied text does not indicate whether the conjecture has been proved or disproved.