Bell's theorem derives empirically testable (in fact, falsifiable) inequalities from seemingly natural assumptions regarding models of EPR correlations. The assumption which is usually rejected is that of Outcome Independence (OI), related to Reichenbach's Common Cause Principle. Some researchers, most notably those of the Budapest school (M. Redei, G. Hofer-Szabo, L. Szabo) try to show that the Principle is not disproved by Bell's Theorem by introducing a new class of models for the EPR correlations, with OI replaced by a weaker condition which is supposedly more in line with Reichenbach's views. It is not immediate whether the assumptions behind the new class of models entail Bell's inequalities. This new condition posits many separate screener systems for pairs of correlated results, in contrast to OI, which requires a common screener system for all pairs of correlated results. We prove that no greater generality is achieved by postulating separate screener systems, and that the assumptions of the new models do entail Bell's inequalities.