No, Guerrilla Games. The engine also already runs on virtually every platform, including the Xbox and PC, but also phones.

Even though Guerilla Games is currently owned by Playstation Studios, they are still effectively an independent studio.

Microsoft, otoh, owns like half a dozen engines, uses none of them, and licenses none of them. The only way I'd see a engine swap happening is if Kojima somehow swung a deal to license the newest version of the Doom engine.

Nearly no chance idTech is being used for anything outside of the few franchises that leverage it, as modern idTech versions are mainly geared towards just those FPS games. id Software has been decimated too in the recent layoffs, and there's questions over the future of idTech