In my own experience, devices will choose the lowest output resolution they can detect (via EDID).
I have an Onkyo amplifier with two HDMI outputs, which I have plugged into a 4K OLED TV, and a 1080p projector.
If the projector is turned on (so the amplifier can read its EDID signal) anything connected to the amp will only output in 1080p. I have to turn off the projector, and then I can output in 4K.
I suppose if you had a HDMI switch with absolute separation of EDID signals (which my Onkyo amp clearly does not have) then perhaps you can leave them both turned on?
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