PoE on Series X is 4k 120 fps. PoE on PS5 is 1080p 60fps. Why is PS5 so gimped?

Hi.
Both versions of the game run in backwards compatibility mode.
PS5 uses PS4 code, which means it targets 1080p with medium textures.
Xbox Series X uses One X code, which targets 4k resolution and uses high resolution textures.
Xbox Series S uses One S code, which is identical to PS4.

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Interesting. Does it matter if I put the console on a lower resolution like 720p, or will the game still render on 1080p only to be downscaled by the console or TV?

The game is not affected by the output resolution of your console/tv. It renders at intended resolution and the image is downscaled/upscaled by the console after that.

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Also, are you working on creating a PS5 enhanced (or native) version, perhaps by utilizing the One X codebase?

Unfortunately I don't have any information to share about this at the moment.

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Final question: does the game support HDR

The game has HDR pipeline, but it does not output HDR image yet. So the answer no for now.

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other output-related features like variable refresh rate?

The game works really well with VRR on Xbox One and Xbox Series. You just need an AMD FreeSync compatible display.

PS4 does not support VRR output. And as far as I'm aware PS5 does not support VRR for back compat titles ☹️