I cannot fully understand the bolded sentence. Thank you in advance.
About the closest any research has come to supporting the tiredness theory is to confirm that adults yawn more often on weekdays than at weekends...
About is not a preposition here. It is an adverb whose meaning is close to "approximately". The research seems to be the closest and the writer is not 100% sure about it. That's why the adverb is used.
There is a noun and relative adverb omitted. That's why it is not easy to parse that part in bold. If you rephrase it to:
About the closest (research) (where or that) any research has come to supporting the tiredness theory is to confirm...
The relative adverb where or that is omitted as there is an antecedent (omitted research) that indicates where the action (any research has come...) happened. The following sentence has the same (or similar) structure.
That house is the closest (house) (where or that) you can find a drinking water in this area.
The easier version would be
There has been no research which was done exclusively to prove the theory (No research has been conducted to prove the theory). However, the closest research that shows the relationship between yawning and tiredness is to confirm that adults yawn more often on weekdays than at weekends, and that school children yawn more frequently in their first year at primary school than they do in kindergarten.