Hypex provides a mute option that might help by asserting this at startup and just before/at shutdown. Manually asserting Amplifier mute using the +5V standby supply and a jumper cable would test the theory. A 555 timer should be enough to automate the delayed turn on and off mute if the amplifier mute function idea is viable.
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Adding an additional amplifier on / standy switch to the chasis that keeps the AC mains on but puts the amplifier into standby with only the 5V standby rail on would work. The switch would simply connect "PS Enable" to "VOUT,STANDBY" or "GND" for on/off states.
Asserting the amplifier mute function with the 5V standby supply (with 250mW idle power) tied to this standby enable switch would automate on/off muting behavior with some additional simple logic circuitry. On 24/7, the standby +5V supply is about 0.2kWh per month.
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I also though about adding a 12V trigger. Adding this should be simple.
Some background on 12V trigger circuits is available from
Elliot Sound Products.
Amazon has some cheap modules that might be suitable.
A second Amazon option.
A master slave power strip is even simpler though.