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I imagine that’s why there are no self-administered tests here.
It's worth noting that that is a report about one brand of one type of, relatively uncommon, test, which has already been reported as being less accurate.
Here's a study on the method of collection generally: "Patient-collected tongue, nasal, and mid-turbinate swabs for SARS-CoV-2 yield equivalent sensitivity to health care worker collected nasopharyngeal swabs" - https://www.medrxiv.org/content/10.1101/2020.04.01.20050005v1
For the lateral flow testing kits (same swab method, but different test to PCR), the BMJ found that the sensitivity (True positive rate) dropped from 73% with healthcare staff to 58% with members of the public.
https://www.bmj.com/content/371/bmj.m4469
PCR test have a much higher sensitivity to start with but presumably there will be a similar drop off due to a relatively poorer swabbing technique by the self-trained public.