Slope estimates of the quantile regression

Question

In the Quantile Regression model $$y_i=x_i^T\beta + \epsilon_i,\ i=1,2,....,n$$ when the error terms are iid then can we expect that the slope estimates for the conditional mean and quantiles to be almost same. If yes why? $$$$Also if the error terms are iid then can conditional quantiles be estimated by adding the corresponding quantile of the error term to the conditional mean obtained by the least squares?