Before revision:  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:  It is thus impossible to construct an estimator that is consistent (or unbiased) for the average treatment effect on the treated (ATT) without either imposing functional form restrictions or imposing assumptions that identify the entire counterfactual distribution of untreated potential outcomes.