Second try - TapaTalk is so buggy!
That looks like a mainboard replacement. If they are refusing warranty that means that they believe that the problem is due to the phone being used in a way that's not covered by the warranty - let's call that "mistreated" for short. So did they say what they think has been done and why they believe this?
That's the first thing to clarify, because once you know that you know what you have to refute. The mainboard is hardly an exposed and vulnerable component, so apart from moisture damage it isn't easy to see how the user damages it without leaving other visible signs. And if I recall correctly you've never rooted, so they can't blame that (which would be nonsense, but not unheard of from a service centre). So you want to know what they claim and what their evidence to support the claim is.
As Marc says, stay calm and polite - going ballistic may be tempting, but isn't likely to be effective.