You've not said how each switcher is connected to each Integra. Do the switchers have dual HDMI outputs, one going to the display and the other to the Integra? Or do the switchers break the audio out of the HDMI and give you an optical audio signal? Or digital coax audio?
Could you go back to the beginning and describe this?
It clearly states that the switcher also has HDMI Outputs that feed the receivers...Take the time to read before you type and ....well, be you.