Uniqueness of asymptotically conical K\"ahler-Ricci flow

We study the uniqueness problem for the K"ahler-Ricci flow with a conical initial condition. Given a complete gradient expanding K"ahler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity--appearing as a K"ahler cone--must coincide with the forward self-similar K"ahler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its K"ahler form remains in the same cohomology class as that of the soliton's self-similar K"ahler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.