Could be that the igpu doesn't properly "hot reset" and that once the vm shuts down it gets stuck with the host.
This theory is easiest to test by checking if a shutdown, remove power, plug in power, startup of the proxmox host allows the existing vm to grab the igpu again.
If that works then a workaround could be to never shutdown the vm.
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If you're passing the iGPU to the VM then the host can't use it if the VM is on.
If your host supports SR IOV, then have a look at this repo: https://github.com/strongtz/i915-sriov-dkms
Installing this on Proxmox and your guest means the guest(s) and host to use the iGPU at the same time.
I'm not sure if it's in mainline somewhere but I had to install this DKMS to get it working.
Just realised your driving the TV with a VM. This probably isn't relevant then.
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