> people, myself included, think there are filaments in our nose (the parts we can see and stick our fingers into) that this is referring to.

The confusion may be due to the fact that olfactory sensory nerves have their cell bodies (with the smell receptors on cillia) in the nose. Like many peripheral nerve cells, these cells also extend directly into the brian via their axons. In this case, those axons extend via the olfactory tract into the olfactory bulb in the brain. These areas (tract and bulb) are where the paper indicates axon damage was found.