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@TJ D I've mentioned this a couple of times where my INT1 powered Wolfbox mirror will drop offline during engine start if I pause the key at position 1 (On). As soon as I turn the key to position 2 (Crank) from position 1 the mirror drops offline. It doesn't drop out if I go from Off to Crank without pausing at On.
I didn't get much response to my posts so was thinking it's a quirk of the mirror. Your observation suggests there is definitely an interruption to the INTx and EXTx outlets during engine start, either a relay changeover or voltage drop as you say.
Not pausing at On might allow the Wolfbox power supply (a 12v to 5v USB-C converter) to buffer enough power to keep the mirror alive. If that's correct it suggests the PWR circuit is deactivated as soon as the key is turned to On which makes sense. Give it a try and see how the Starlink handles it.