Neither, as it happens. To sport one's oak is a rather dated expression, mainly from the universities of Oxford and Cambridge. College rooms of the older sort usually had two doors, an inner one for ordinary use and an outer, more massive wooden door, called the oak , which was normally folded back against the outside wall. By convention, if you closed the outer door you indicated that you wanted to be left undisturbed, say because you were giving a tutorial.
Sport here is an old use of the verb, meaning that one was exhibiting or showing something.
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