  It is well known that the effect of quantum nonlocality, as witnessed by
violation of a Bell inequality, can be observed even when relaxing the
assumption of measurement independence, i.e. allowing for the source to be
partially correlated with the choices of measurement settings. But what is the
minimal amount of measurement independence needed for observing quantum
nonlocality? Here we explore this question and consider models with strong
measurement-dependent locality, where measurement choices can be perfectly
determined in almost all rounds of the Bell test. Yet, we show that quantum
nonlocality can still be observed in this scenario, which we conjecture is
minimal within the framework we use. We also discuss potential applications in
randomness amplification.
