There are a variety of approaches to handle informative prior distributions when they disagree sharply with the data. I want to summarize a few papers that present these approaches, mostly so I can keep better track of them. You may find this summary helpful as well.
Kozubowski TJ, Alhamzawi R, Yu K. Power Prior Elicitation in Bayesian Quantile Regression. Journal of Probability and Statistics, 2010-12-01, doi: 10.1155/2011/874907. Available in html format or pdf format