The 5V output on the HDMI port is missing the required overcurrent protection. ("An HDMI Source shall have +5V Power signal over-current protection of no more than 0.5A. All HDMI Sources shall be able to supply a minimum of 55 mA to the +5V Power pin." Also relevant: "No damage to the HDMI Source or Sink can result from the shorting of any combination of signals on any connector.")
The HDMI-side of the level shifter should obviously use H5V as supply, not VDD_5V or any other 5V supply.
I get the feeling that HDMI hotplug detection can't possibly work right now, since it needs to distinguish between HDMI_HPD being pulled up to H5V via 1KΩ (sink connected) versus HDMI_HPD being floating (nothing connected).
This will be fixed in rev A2.
For rev A1, you'll need to have the monitor plugged in when booting BeagleBone AI or manually set the resolution.
jadonk
commented
4 years ago 
End user HDMI feedback:
This will be fixed in rev A2.
For rev A1, you'll need to have the monitor plugged in when booting BeagleBone AI or manually set the resolution.