I'd make sure there's appropriate `surface=*`, `smoothness=*`, `tracktype=*` etc. tags, but I'd treat this as a `highway=path`. Just because something is part of a cycle route doesn't mean it's a cycleway, in the same way some parts of Flemish cycle highways are tagged as `highway=track` because they're reused agricultural tracks.

More specifically, I agree with the currently mapped situation, where https://www.openstreetmap.org/way/554303353 is a path and the nearby sections are cycleways. That's really how it felt on the ground for me.

So essentially, I suggest to close this note.