I'm running FLAC files via the VOX app, 0,2m lightning cable to Fiio L9 LOD cable to Fiio A3 amp with B&W P7 headphones - I don't find any distortion in the highs. I also run podcasts via the Downcast app, and youtube via the TubeX app. If anything, it sounds better to me because the phone's amp's job is now being handled by the Fiio using the lightning to 30-pin. Perhaps it's a problem with your phone's lightning port? Or the amp/output you're using? Have you tried testing the adapter against another lightning port? Different amp? Headphones?

Yes, I'm facing the same problem since iOS9. The sound quality was great during iOS8 with the lightning adapter. What a shame that now have to use back my old iPod Mini 3rd Generation which now sound better than my iPad Mini 2.

Hope that Apple will rectify this issue in the upcoming new iOS update.

I found the issue, it's Siri. Turn off siri in setting and the sound quality is restored. You can go into settings while music is playing and turn Siri off and here the quality improve. I have the iPad Pro 9.7 and the iPhone 6s and neither device has this issue.

I would try to update to the latest iOS 9.3.4 and see if that works for you. It seems like other users are experiencing this issue. I'm going to do a little research and see if I can find something for you.