According to the paper, both words coexisted during Old English. It is less concerned about how they got there and more interested in how with took over once they were both established.
Word meaning can be fluid though. Like Wrist in English and Vrist in Swedish mean completely different body parts - despite the words obviously being related.
English has a special way of changing meaning - so words like "meat", "deer", "foul" and "hound" originally meant something like any "food", any "animal", any "bird" and any "dog".