  We study $n$-dimensional Ricci flows with non-negative Ricci curvature where
the curvature is pointwise controlled by the scalar curvature and bounded by
$C/t$, starting at metric cones which are Reifenberg outside the tip. We show
that any such flow behaves like a self-similar solution up to an exponential
error in time. As an application, we show that smooth $n$-dimensional complete
non-compact Riemannian manifolds which are uniformly PIC1-pinched, with
positive asymptotic volume ratio, are Euclidean. This confirms a higher
dimensional version of a conjecture of Hamilton and Lott under the assumption
of non-collapsing. It also yields a new and more direct proof of the original
conjecture of Hamilton and Lott in three dimensions.
