I don't see how that can work for the push packet itself, cause I thought that's specially handled by some low-power hardware on the phone while the main parts

I don't see how that can work for the push packet itself, cause I thought that's specially handled by some low-power hardware on the phone while the main parts are shut off. Unless that hardware is also managing the VPN connection, which I doubt. So if there's no always-on hardware maintaining that VPN connection, probably the phone is going to wake up without it. And even if it auto-reconnects, it'll probably load stuff before it's connected to the VPN.