Are reference points merely lagged beliefs over probabilities?

What explains the mixed evidence from laboratory tests of Kőszegi and Rabin’s (2006 and later) model of expectations-based reference-dependent preferences? We investigate one hypothesis: to become (behavior-affecting) reference points, probability beliefs have to sink in—being merely lagged, as the theory requires, is not sufficient. Past experiments with conflicting findings exogenously endowed subjects with beliefs that were equally lagged, but possibly unequally sunk-in. In four experiments, whose designs replicate past KR-nonsupporting experiments, we add new sink-in manipulations that endow individuals with additional, visual/physical probability impressions. Our findings are more KR-supporting in an endowment-effect setting but not in an effort-provision setting.