TRMOTOR is %100 owned by TAI, and TEI is also %51 owned by TAI. TRMotor is a design bureau, and TEI is a factory. TEI design team (biggest), the TRMotor design team(TRMotor recruited engineers), and the Ivchenko Progress design team (their position is similar to BAE input in KAAN) work under TRMotor that way all of the design capabilities came together under a single structure to unify design efforts. It seems like this approach accelerated the engine project.
IMHO RR missed its chance. Bad for them they will lose billions. Because I expect Kaan to sell like hotcakes. There won't be any Kaan alternative on sale in 2030s. KAAN will be the only 5th gen fighter available at that size and kinematic capability with stealth and 5th-generation sensor suite. Tempest won't be around. French FCAS won't be around. AMCA won't be equal as it is a smaller fighter. F-35 will also be smaller and it comes with a lot of strings attached.