Randomizing the Number of Centers in k-means++

The $k$-means++ algorithm is a standard and widely used seeding method for $k$-means clustering, but for a fixed number $k$ of centers its worst-case expected approximation ratio is
$\Theta(\log k)$. We consider the same algorithm when an adversary first fixes the dataset and some $K$; the number of centers $k$ is then chosen uniformly from $\{K,\ldots,2K-1\}$.
We prove that $k$-means++ is an $O(1)$-approximation with constant probability in this budget-smoothed setup.