K-means random initialization: two centers at same point

Question

I'm wondering how k-means deals with randomly initializing two centers (for two distinct observations x_1 and x_2, but with the same spatial localization) that are at the same point? I'm planning to cluster a bunch of observations I have, but I would ideally like to keep observations that have the same spatial localization in the same cluster. In the picture the two points at the bottom could have been randomly assigned to be two centers.

I've seen a similar answer asked here: In k-means, can two initial random centeroids be same?, but I'm wondering more how k-means handles situations like this algorithmically.

Some background in sklearn: As I understand it, in python you can initialize your data using a random initialization, whereupon I don't think there is any restraint on different centers being in the same point, and also there is the kmeans++ initialization, where my question does not apply, because kmeans++ finds centers "spread apart".

Answer 1

Quoting the R documentation for kmeans

 centers: either the number of clusters, say k, or a set of initial
          (distinct) cluster centres.  If a number, a random set of
          (distinct) rows in ‘x’ is chosen as the initial centres.

If I read this correctly, if two observations are perfectly overlapping they cannot be two centers because the algorithm needs distinct rows.

If, instead, they are nearly overlapping, they should be handled like any other observation. In this case, the documentation reads:

In rare cases, when some of the points (rows of ‘x’) are extremely close, the
algorithm may not converge in the “Quick-Transfer” stage, signalling a warning
(and returning ‘ifault = 4’).  Slight rounding of the data may be advisable in
that case.

I suppose other implementation of kmeans follow the same logic?