There is NOT enough data to really evaluate the problem. It could be that your action is too high and you are squeezing the strings too much, causing them to bend out of tune. You could also be bending them slightly when you play.
It may not be the neck. If it is the neck then you must be cranking on it in an unnatural way when you play. This could be ...
Based upon that audio, I can hear all the strings in the chord shifting, and by a lot.
Sounds like it might a serious issue, like a crack opening and closing.
That's a pretty expensive set neck guitar. Take it into a guitar shop for a look by someone else, preferably their in house tech.
There are things you can't tell by description alone, and the class "What's wrong with my instrument?" is solidly in that category. Stating that, there are a few hopefully helpful things I can say.
Talk To Your Local Tech/Guitar Shop - That person has been dealing with instruments for a while and can tell if the instrument is in bad shape or if it'...