Before revision:  This assumption necessarily holds when treatment is (as if ) randomly assigned, but will often be implausible in settings where randomization fails. We show further that the average treatment effect on the treated (ATT) is identified regardless of functional form if and only if the entire distribution of untreated outcomes is identified for the treated group. It is thus impossible to construct an estimator that is consistent (or unbiased) for the ATT regardless of functional form unless one imposes assumptions that identify the entire counterfactual distribution of untreated potential outcomes.
After revision:  This assumption necessarily holds when treatment is (as if ) randomly assigned, (ii) the distribution of potential outcomes is stationary, or (iii) treatment is as-if randomly assigned among a subset of the population and the remainder of the population has stationary potential outcomes. We show further that it is impossible to construct an estimator that is consistent (or unbiased) for the ATT regardless of functional form unless one imposes assumptions that identify the entire counterfactual distribution of untreated potential outcomes.