brainteaser debunked

There is an old brainteaser that's been around a long time. I figured it out when I was a kid and I read it in the back of Discover Magazine. There's even a variation in the movie Labyrinth. It goes something like this:

You're in a room with two doors. There's a guard at each door. One door is the exit, but behind the other door is something that will kill you. You're told that one guard always tells the truth and the other guard always lies. You don't know which guard is which. You are allowed to ask one question to either of the guards to determine which door is the exit. What question should you ask?

The generally accepted answer is:

Ask either guard what door the other guard would say is the exit, then choose the opposite door.
If you ask the guard who always tells the truth, he knows the other guard would lie, so he'll point you to the door leading to death. If you ask the guard who always lies, he knows the other guard would truthfully show you the exit, so he'll lie and point you to the door leading to death.

There is a problem with this solution. If the scenario were truly a binary one (Door #1 or Door #2) then asking the question as phrased would work. However, the liar is a variable you can't predict. As a result, there are THREE possible answers that either guard could give: "Door #1", "Door #2" or "I don't know".

If you asked the liar what the other guard would say, he, being a liar, could answer "I don't know"

If you asked the honest guard what the other guard would say, he honestly doesn't know because he knows the liar could answer with the wrong door or just simply answer "I don't know". So the honest guard would also answer the same.

I don't know that there is a true solution to this brain teaser, but I do know that the generally accepted answer is wrong.