Spoiler Alert
I argued that we were not testing for "identicalness of coffees" we were testing for "presence of stick". In fact, the two cups did not have to be identical, except for stirring method. One could have been coffee, the other tea. It was the unpleasant taste of the stick we were testing for, not whether or not the two cups were different. Even if there were unpleasant oils in the wood, it was irrelevant to the purpose of the experiment if they could not be detected by their taste.
But I was never able to convince my friend. What do you think?