How to trigger power button on appliance? Mini relay?

I have an air purifier that I would like to run automatically before bed time. If I could just use a zigbee power outlet/plug, it would be easy. Sadly, it does not turn on when it gets power. I need some sort of relay to trigger the power button on the device in addition to start giving it juice.

I have not dismantled it yet to see how it’s wired, but i would guess, the power button momentarily closes a loop when pressed, triggering a relay of sorts powering the thing up. If that’s the case, a relay that momentarily closes the loop the power button is connected to, should suffice. esp32 with a relay seems overkill for this, and would probably also need to be powered, requiring me to dig more into the unit than just the button to hook that up, which I’d like to avoid.

I assume there is must be some voltage running through the power button loop (when closed) that could be used for something. Does anyone know of a (preferably very small) relay that would trigger the circuit for 100ms or so in a use case such as this? I’m thinking I can’t be the first one that wants to control an appliance needing a button press after power is supplied. I’d like to parallel-connect the power button and the mini-relay, so that both can close the loop, as the power button also controls speed/power level and would be nice to still have.