Context: quantile regression with a binary predictor, but this question can be generalized to other quantile regression model structures and possibly splines/adaptive models.
In quantile regression there appears to be repeated testing for selected quant values in regards to the estimated dependent variable. I have not seen mention of correcting alpha levels for false discovery given the repeated testing. Is this a concern and what have others seen in this area?
EDIT: Example figure below (right pane) where estimates and ttests are calculated for quantile = (0.05 0.25 0.50, 0.75, 0.9). So, a ttest gets calculated for these 5 quant values (e.g., quantiile: 0.05, p-value: <0.9999,...,quantile: 0.9; pvalue: 0.0314)