The answer is no, because it's easy to construct an example that is sensitive to the initial centroid guesses. For example, suppose your data has some points with values close to 1, a similar-sized bunch of values close to 5, and a single point at 3. For k=2, whichever initial centroid is closest to 3 will initially claim the point at 3, and will retain it in the next iteration and the algorithm will terminate.
That's enough to answer the question, but I'd also suspect that in any situation where the "actual" number of clusters differs from the chosen k, you'd get this type of behavior as well. For instance, if there are "really" 3 clusters but k=2, the closest initial centroid guess to the middle cluster will grab and keep the biggest number of those, maybe all of them because the initial "losing" centroid will keep moving towards the extreme, and the initial "winning" centroid will move closer to the middle.
So, your data may naturally always converge to the same answer, but that's not necessarily always the case.