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This, Peter Pedant, is why I'm taking this slightly different approach. They won't 'sing' if that's what I call it. But they will join in with some of the cheesiest, daftest songs the boss puts on that Christmas playlist. Often it's just to have a laugh, and they do stupid voices or make up rude words because, as you suggest, that doesn't compromise their perception of manliness.
Just a couple of weeks ago they decided to have a 'Good Life' stint and they were all going around pretending to be Richard Briers whistling that little 'Somewhere Over the Rainbow' that he does and wondering what the words were - probably so that they could make up something daft to replace them with. So they will do it, but just not if it's presented as anything formal or serious.