Even lossy compression, I don't see how it could work technically given the transfer methods allowed.
BUT! The diagram you showed does make me think of one way they could be doing it.
It is very inexpensive to convert that HDMI into a 480i analog composite video signal. Send that as a balanced line a great distance. Then cheaply upscale it back to a 1080p digital HDMI signal.

That is a VERY dirty thing to do. It would look like crap. But it would work. And seeing what look like transformers on the output stages, is a strong indication that might be exactly what they are doing.