Here’s a very technical paper that studied nose vs mouth vs combined nose-and-mouth breathing:

www.ncbi.nlm.nih.gov/pmc/articles/PMC7455204/

I confess it was over my head and I just skimmed it. But the conclusion says, “The high filtration efficiency of the nasal cavity together with its efficient clearance mechanisms lead to the recommendation to prefer the nose breathing over combined or mouth breathing.”

The conclusion also says, “There is general scientific agreement that lower airways are more vulnerable to severe infections” and “From this point of view, the nasal inhalation is preferential because it significantly reduces the number of particles penetrating to lower airways.” I’d guess that means that shallow breaths are probably preferable, but you’d need to read the article to confirm that.