This paper examines the superiority-claim of the GARCH based measure in resolving the idiosyncratic risk-return puzzle using Australian data. The least squares and the quantile regressions of stock-returns on lagged idiosyncratic-volatility estimated from daily data using two measures (including GARCH) fail to support such claim. The quantile regression estimation reveals the risk-return relationship to be quantile dependent; it is parabolic but significant only at the extreme quantiles. The parabolic-form is convex (concave) at the lower (upper) quantiles of the returns conditional distribution. This changing relationship-form reflects uncertainty in predicting returns. Moreover, the idiosyncratic risk-return puzzle is a model specification problem.