This paper gets ripped into in my favorite podcast, TWiV (This Week in Virology)!
Essentially, yes, neomycin in the nose, if timed perfectly, can activate the innate immune system, but en mass this practice would cause the spread of antibiotic resistance.
If it turned out to work great in practice and people started using it en masse, the benefits would greatly outweigh the costs IMO, particularly if it snuffed out Covid and/or the flu. If it ends up being a niche thing, I doubt it would bite into the resistance numbers.
Besides, we're already spiraling down the resistance chasm with antibacterial soaps, stuffing cattle with antibiotics, overprescribing, and so on.
Besides, we're already spiraling down the resistance chasm with antibacterial soaps, stuffing cattle with antibiotics, overprescribing, and so on.