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

A steady gradient Kähler–Ricci soliton is a complete Kähler manifold with a metric satisfying the steady gradient Ricci soliton equation. Cao's conjecture. Every complete steady gradient Kähler–Ricci soliton with positive curvature on $\mathbb{C}^n$ must be isometric, up to scaling, to Cao's $U(n)$-invariant steady gradient Kähler–Ricci soliton on $\mathbb{C}^n$. Cao constructed this soliton, which has strictly positive sectional curvature; the conjecture asserts its uniqueness among complete positively curved steady gradient Kähler–Ricci solitons on $\mathbb{C}^n$.