What might the true sensitivity be for lateral flow Covid-19 tests?

Disclaimer: I am in absolutely no way an expert on Covid-19 or tests for it. This post was motivated from undergraduate teaching for medical statistics, and probably makes assumptions which are implausible.

This morning I spoke to my undergraduate medical statistics students briefly about this piece in the BMJ published a few days ago raising concerns about the possible low test sensitivity of the rapid lateral flow tests which are being used in various contexts currently in the UK. This piece cites a UK government document which says concerning the lateral flow tests that:

Results of the PHE and Oxford University Innova evaluation show it has an overall analytical sensitivity of 76.8% for all PCR-positive individuals but detects over 90% of individuals with high viral loads

https://www.gov.uk/government/publications/community-testing-explainer/community-testing-a-guide-for-local-delivery#what-the-community-testing-programme-is

and that

In field evaluations, such as Liverpool, these tests still perform effectively and detect at least 50% of all PCR positive individuals and more than 70% of individuals with higher viral loads in both symptomatic and asymptomatic individuals.

https://www.gov.uk/government/publications/community-testing-explainer/community-testing-a-guide-for-local-delivery#what-the-community-testing-programme-is

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