Before revision:  Since differentiation of the latter has implications for nonparametric (point) identification and estimation of the conditional quantile function , our second test is designed to detect misspecification . Using only individuals with propensity score close to one, this  test relies on an `identification at infinity' argument, but accommodates cases of irregular identification.
After revision:  Since differentiation of the latter has implications for nonparametric (point) identification and estimation of the conditional quantile function (s), we suggest a second test to identify whether the cause for rejection at the first stage was solely due to selection or not . Using only individuals with propensity score close to one, this test relies on an `identification at infinity' argument, but accommodates cases of irregular identification.