> There's a lot of evidence that exercising is healthy and helps people improve their mental state so that analogy doesn't work.

There's actually a lot of research out there. Did you try to search for any? It's not scepticism to compare something you cannot believe because you haven't bothered to investigate, with well known falsities such as crystal healing.

Not only does the research support the existence of a nasal cycle where one nostril is dominant at different times, it also supports a different pathway to the brain and a different neurological response. On top of that, psychological research shows links between breathing patterns and emotional response, in both directions, and then there's the wealth of meditative research. This stuff isn't controversial.

From [Taste and Smell]:

> The FN model may also explain the observation by Sobel et al. [36] that, when performing an odor threshold test, humans sniff longer when using the nostril with the lower flow rate. (Subjects usually have different flow rates in their two nostrils because of cyclic changes in the size of their nasal cavities.)

From [Nature]:

> The flow of air is greater into one nostril than into the other because there is a slight turbinate swelling in one. The nostril that takes in more air switches from the left to the right one and back again every few hours

From [Stanford News]:

> Michael Leon, a professor of neurobiology and behavior at the University of California at Irvine said… [the] data suggest, that the olfactory system maximizes the ability of its distributed receptor neurons to encode differentially absorbing odors.

From [Measuring and Characterizing the Human Nasal Cycle]:

> Nasal airflow is greater in one nostril than in the other because of transient asymmetric nasal passage obstruction by erectile tissue. The extent of obstruction alternates across nostrils with periodicity referred to as the nasal cycle. The nasal cycle is related to autonomic arousal and is indicative of asymmetry in brain function. Moreover, alterations in nasal cycle periodicity have been linked to various diseases.

From [Hemispheric lateralization in the processing of odor pleasantness versus odor names]:

> These findings are consistent with previously demonstrated neural laterality in the processing of olfaction, emotion and language, and suggest that a local and functional convergence may exist between olfaction and emotional processing.

From [Respiratory feedback in the generation of emotion]:

> This article reports two studies investigating the relationship between emotional feelings and respiration. In the first study, participants were asked to produce an emotion of either joy, anger, fear or sadness and to describe the breathing pattern that fit best with the generated emotion. Results revealed that breathing patterns reported during voluntary production of emotion were (a) comparable to those objectively recorded in psychophysiological experiments on emotion arousal, (b) consistently similar across individuals, and (c) clearly differentiated among joy, anger, fear, and sadness. A second study used breathing instructions based on Study 1’s results to investigate the impact of the manipulation of respiration on emotional feeling state. A cover story was used so that participants could not guess the actual purpose of the study. This manipulation produced significant emotional feeling states that were differentiated according to the type of breathing pattern. The implications of these findings for emotion theories based on peripheral feedback and for emotion regulation are discussed.

I could go on.

As to meditative practices, there's been extensive research for nigh on 50 years now, and for the past 20 (at least, as far as I'm aware) they've been using fMRI scanners to provide objective results[Oser] on the link between meditation as a practice and mental state.

You just have to look.

[Taste and Smell] https://books.google.com/books?id=fuxS-p6bpuwC&pg=PA12

[Nature] https://www.nature.com/articles/46944

[Measuring and Characterizing the Human Nasal Cycle] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5053491/

[Stanford News] https://news.stanford.edu/news/1999/november10/smell-1110.ht...

[Hemispheric lateralization in the processing of odor pleasantness versus odor names] https://www.researchgate.net/publication/12712833_Hemispheri...

[Respiratory feedback in the generation of emotion] https://www.tandfonline.com/doi/pdf/10.1080/0269993014300039...

[Oser] https://www.lionsroar.com/the-lama-in-the-lab/ I include this as it's a personal favourite of mine and covered more extensively in Daniel Golman's book, Destructive Emotions.