Cao's classification conjecture for positively curved steady Kähler–Ricci solitons

Let $(M,J,g,X)$ be a simply connected complete steady gradient Kähler–Ricci soliton, meaning that $g$ is a complete Kähler metric and $X$ is a vector field generating the soliton. Suppose that $g$ has positive sectional curvature. Cao's conjecture. $(M,J,g,X)$ is isometric to the ${\rm U}(n)$-invariant steady gradient Kähler–Ricci soliton on $\mathbb{C}^n$. Cao's conjecture proposes a classification of complete positively curved steady gradient Kähler–Ricci solitons; the source provides no evidence of a resolution, so its status remains open.