There are a lot of similar error messages about one of the HDMI/DP audio outputs, but a physical connection problem would also be present on any other OS
!!ALSA/HDA dmesg
snd_hda_codec_hdmi hdaudioC0D2: HDMI: pin nid 5 not registered
This is not like most ALSA problems on other systems that usually relate to the analog inputs and outputs
I suspect this may be similar to issues from last year with Intel laptops using a similar audio codec - Intel Corporation Cannon Lake PCH cAVS that had problems with digital microphones and HDMI audio
I suggest you try using a later kernel to see if the sof-firmware drivers will help get the digital outputs working again
Use Manjaro Settings Manager > Kernel GUI to install kernel v5.10.23
or use this command
sudo mhwd-kernel --install linux510
Then reboot system and check if HDMI outputs are now detected
If that does not work for you, try this command to force ALSA to use the ‘legacy’ driver snd_hda_intel
echo 'options snd-intel-dspcfg dsp_driver=1' | sudo tee -a /etc/modprobe.d/alsa-hda.conf
then reboot system again to make sure this option is loaded