I know it has been a while since you asked but I would rather it be technically inaccurate in order to appear closer to how it looks on screen/interview footage. I know that on screen they usually use a lot of filters and post processing so I think if you aim for the appearance of the shirt in the interview footage it would be a good balance. I wouldn't want it to look quite as dark as it does in some scenes of the movie.
Also, I am sure you have considered, that the shirt appearing on the costume displays is not the same shirt that hugh wore in the movie/interviews. It is possible they did more work on his shirt to get the washed out appearance.