Can the university cut the notice period short?
I suspect (but have no legal standing) that they are obligated to pay you through the notice period even if they ask you to leave early. 

But it also seems like they still are bound by a six month notice period rather than suddenly getting a shorter one because of your action. 

But suspect and seems are pretty dangerous to rely on from someone who is just speculating. And, no one here can give you legal advice even with full information. We can only guess and hope we are correct. But you are the one on the line here. 

I worry, also, that you have an adversarial attitude about HR. I suspect (that word again) that they are legally obligated to give you proper advice. They probably don't require all of the details of your situation to do so.