I can only vouch for up to 5 minutes at 4k 30fps, which I've only done a couple times, and not in direct sunlight. No issues as far as I can remember. I don't even remember it getting hot, but then again I generally don't notice if/when my Air gets hot since most of the heat comes from the plateau which I don't touch often. It usually has to be extremely hot for me to notice, so I don't think it got extremely hot while shooting the video. But I wouldn't fully trust my memory, you should try to get more confirmation.
But I'll say I'd think there would be more noise online if the Air overheated dramatically in only 10 minutes of video recording. Then again maybe that much recording is not common.