My understanding from careful study of Bells theorem is that there may be a deep subtlety that is not ruled out, but is probably very minimally studied in the literature. However the literature is vast, and ruling out that it is not studied at all seems questionable verging on unlikely.
- suppose that the hidden variable(s) control whether the particle is detected. is this ruled out by bells thm?
- if not, is that postulate enough to lead to a consistent theory, compatible with the known QM formalism?
There is a lot of analysis of "biased vs unbiased sampling" in some of the literature, but it seems reasonable to me to argue that its anthropomorphic thinking to talk about it in those terms, and maybe one should just start from the (actually very reasonable?) assumption that the only possible explanation is that the hidden variable(s) are controlling detection of the particle (ie presence vs absence). There are also some recently declared "loophole free bell tests", but I am also seriously wondering if those are not ruling out this above scenario either.
There may even be circumstantial evidence in the form of many, many experiments that have to carefully consider/ account for detection efficiency which is not really directly considered in bells original (thought) experiments. Maybe "detection efficiency" is really optical/ technical shorthand for "hidden variables controlling detection."
The above scenario is briefly discussed in the book The infamous boundary by Wick and was supposedly considered decades in the past and labled "a conspiracy theory" but again that seems like possibly biased thinking to me.