Occam’s Razor : look for simpler reasons. Remember too, there are atmosperic interactions here. To verify this 4 minute delay you need space based gamma ray telescopes free from atmospheric effects. I’m sure that many will look at this and find all sorts of reasons to “save” relativity. I’ll ponder this as a good conundrum to get off to sleep tonight.
4 minutes is an eternity for any explanation based on atmospheric effects local to the Earth. It takes just 8 minutes for light to reach the Earth from the Sun.
On the other hand, .4 minutes is an extremely short period of time, relative to the total distance travelled by those gamma rays.
But there’s no need to draw any final conclusions right away. This result can and should be examined and debated from every angle, probably for years.