Cure = debatable word.
But N=1, yes. I had horrible allergies and sinus problems for years, but after going paleo/primal/ancestral/ whatever I no longer suffered like I had previously. The symptoms described in the article are due to inflammation, no? Reduce the inflammation = less/ no symptoms.
However, re: the article -- the authors don't include pollution factors -- my reading tells me it was not only pollens I was reacting to, but environmental pollution. I still experienced some symptoms when I was living not only in a high-pollen, Santa-Ana winds area (Riverside, Ca), but a high-pollution factor area. But I was not fully paleo/ etc then. Since I have moved out of state to a almost zero-pollution area, I have experienced no symptoms at all.
Hoped this helped to answer your question.