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Does Looping Counter qualify as kolmogorov-complexity?

Lundin just suggested in a [comment under another question](https://codegolf.codidact.com/comments/thread/6229#comment-17347) that I tag the challenge [Looping Counter](https://codegolf.codidact.com/posts/286280) as [kolmogorov-complexity](https://codegolf.codidact.com/categories/49/tags/4275).

Now I'm not sure if it actually qualifies for that tag, for the following reasons:

  * It is an infinite loop producing an infinite output, unlike the existing challenges tagged that way which all produce a finite output.

  * The task allows for an arbitrary printable character to be used (and some solutions take advantage of that). Thus there are many possible outputs (one for each printable character).

So should I add the tag to that challenge? More generally, which, if any, of the above points are relevant on whether a challenge qualifies as kolmogorov-complexity?