Hi, sorry for not getting to you earlier, your question is relevant and well written.
Unfortunately this seems to me like a very tough problem that lies mostly in the design of the data collection. If you had at least some data on how disease looks like for people with lower test scores, the whole thing would be reasonably easy. Even incoroporating such data from different studies/populations/… might be easier than trying to answer the question from the data you have. Or if you have some idea on what the expected total prevalence in your population would be (which we could then use to infer the number of cases we missed).
Without additional data, I think you will need to bring a lot of untestable assumptions to the table to be able to do anything. You basically need to somehow extrapolate the relationship between test and disease (and potentially test2) from cases where test >= 0.88 to the rest of the cases. And extrapolations are tough. E.g. even if there was a clean linear relationship between log odds of disease and test for high test values, how can we justify extrapolating this line all the way towards say test = 0.25?
If you can consult the knowledge of the particular disease and test to constrain the possible form of the relationships that might help.
It seems to me that your case is very similar to that discussed at: MICE missing values on the response If that is right than the best you can do is to fit the model to the cases where you have the response and hope they generalize, but as I said above, I don’t think such hope would be easy to justify.
Best of luck with your model!