Hp office jet pro series

When you say "chronically light up", what do you mean?   Do you mean the control panel on the printer itself?  

Once a WiFi connected printer is initially connected to the network, said printer will always be connected to that network if it, the printer, is turned on.    

I had to connect a new Brother monochrome laser printer for my folks, and it's the first time I've ever connected a printer via WiFi.  Every time they power up their system (HughesNet and computer) their computer gives them "printer connection failure" notices if the printer isn't turned on, as well.  I've searched and searched through the settings and I can't find anything, thus far, that will stop this.  They use their computer nearly every day, yet print something perhaps once a month.  It's stupid that they need to have the printer on all the time.   :(