Yes, this is the specification for THEIR I2S port.
The problem is that I2S doesn't include a standardized specification. It does NOT have a standard cable, connection, or pinout from device to device to device and was never meant to act as a externally cabled means of audio transport. It was designed for integrated internal (proprietary) connectivity.
So, you would need a I2S output connection, then figure out all the pinouts, then connect them properly. If you are lucky, you would get an audio source which uses I2S and has the exact same pinout configuration as this device uses, then a simply HDMI cable should work fine. But, because there is NOT a standard pinout, there is no promise that the HDMI cable would work and you might need to cut the HDMI cable and splice all the connectors in the middle to make a connection work.
In simple terms. I2S is stupid. It's just flat out mental. Any communications standard should start with a stanardized interface so that devices can communicate. This is just not part of the way this was designed. Stupid.
There are some sites which discuss it, but Wikipedia just covers how this doesn't have any standardization whatsoever.
Which makes it stupid.
If you go this route, you may not be entirely on your own, but because it is VERY device specific, you would need a lot of time with each specific device to ensure all the pinouts are compatible and a standard HDMI cable will work. Since each device is different, there is no 'standard cable' which anyone can tell you to buy and they don't exist. There is NO I2S cable in the world, since I2S has no standard to build a cable for.
I have no idea what harm would come if you connected a I2S HDMI connection to a standard HDMI input. You could break that input... I wouldn't try it.
en.wikipedia.org
Specifically:
This is a non-consumer format. It is stupid to put it as a connection on any device. Just stupid. It is confusing, as you are finding out, and pointless.