Say we fitted quantile regression models with a set of quantiles between 0 and 1 and a linear regression (i.e. mean regression) to a same dataset with the same set of covariates. In terms of the estimate for regression coefficient for a specific covariate, say x, we will get nine estimates from the quantile regression corresponding to each of the nine quantiles and only one estimate from the mean regression. If we plot those estimates in the same plot, with y-axis represents the values of the estimate and x-aixs for the quantiles, we should get a horizontal line for the estimate from mean regression and a zigged line connecting the estimates from quantile regression.
My question is if these two lines must cross at some point (i.e. some quantile) in every case or is it possible for them not to cross with each other for all possible quantiles?
Is the conclusion consistent in the setting of linear quantile mixed model and linear mixed model?
edit @05/13: sample plot inserted