It really could be anything right now....
DI engines are kind of known to carbon cake the intake valves.
I have 3-4 DI cars here and have never had the carbon problem that I'm aware of.
I guess we are offering ideas on what it could be.
Pretty sure you will need a scan tool to see any engine data....one of the parameters would show the fuel pressure. No way to hook up a gauge on a high pressure system. Pretty sure you will not find a fuel filter.
You might find this link helpful..