What if you plugged in a "DVI to…[some other standard]" adapter. It may be that the DVI port activates in some way when something is plugged in, even if that something is off, not powered on. By this thinking you would not have to connect the other end of the adapter to anything; you just want something that plugs into the DVI port. Make sure whatever you get can be returned because this is all a wild guess on my part.
