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Thank you for your interest in the Intel® Edison Breakout Board.
The state of the GPIOs is a topic that has been discussed several times in this community, the biggest issue being the unpredictability of said pins during start up. The previous has lead to developers creating their own images to counteract the fact that the digital pins are shared by different interfaces through the multiplexing process.
But since your inquiry is related to the state of the GPIOs during shutdown, it seems that, as you already mentioned, inverting the logic of your design is the way to go in order to respond to the pull-down generated by the compute module.
If you have any other question or update, don’t hesitate to contact us.
Have a nice day.