A note on Wu-Zheng's splitting conjecture

Cao's splitting theorem says that for any complete Kähler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\mathbb{C}^k\times (N,h(t))$, where $(N,h(t))$ is a Kähler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$, where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu and Zheng when curvature is assumed to be bounded.