Part of the claimed innovation here is that the thing sticks directly into the user's nose. If it seals well, it's plausible that there's very little wasted air, which would account for the factor of roughly five.

(Also, you used the formula for isothermal compression, I think. A tiny little device like they showed would probably move air quickly enough through its fan that it would be closer to adiabatic. This doesn't matter much for the small pressure change involved.)