  In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient
K\"ahler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady
gradient K\"ahler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is
necessarily $U(n)$-invariant (and hence unique up to scaling). Recently,
Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we
construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant,
complete steady gradient K\"ahler-Ricci solitons with strictly positive
curvature operator on real $(1,\,1)$-forms (in particular, with strictly
positive sectional curvature) on $\mathbb{C}^{n}$ for $n\geq3$, thereby
answering Cao's question in the negative for $n\geq3$. This family of steady
Ricci solitons interpolates between Cao's $U(n)$-invariant steady
K\"ahler-Ricci soliton and the product of the cigar soliton and Cao's
$U(n-1)$-invariant steady K\"ahler-Ricci soliton. This provides the K\"ahler
analog of the Riemannian flying wings construction of Lai. In the process of
the proof, we also demonstrate that the almost diameter rigidity of
$\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the
curvature operator is bounded below by $2$ on real $(1,\,1)$-forms.
