Yup. A quick search suggests HDMI runs at clock speeds of 25MHz to 340MHz. Not running that over a relay! Also HDMI is many pins, so you would need a low impeda

Yup. A quick search suggests HDMI runs at clock speeds of 25MHz to 340MHz. Not running that over a relay! Also HDMI is many pins, so you would need a low impedance high pole count relay even if it was possible, and such a relay plus the device wrapped around it plus some profit margin means you're not making it any cheaper.