I don't know what the background is to why the BBC might be claiming that the 'ozone hole' is repairing itself. My understanding of its behaviour in recent years is that the hole (which is a thinning of the ozone layer rather than an actual hole) isn't really going away. For example, this blog post from WUWT, written only a year ago in 2015, provides a graph of how the ozone hole size is varying with time, and the graph does not suggest that a significant repair is occurring:
My guess would be that there might be several organisations who are attempting to measure the size of the ozone hole, and there could be more than one ozone hole area versus time dataset. It may be that in one of these datasets the ozone hole is appearing to reduce in size a bit more than it is in others.
In regard to the 'conclusive paper' that the climate alarmists are relying on, my guess is that the nearest equivalent of this idea for them would be the IPCC reports, which is of course a series of reports written by themselves. The latest IPCC report was issued in 2014, and as I remember it the BBC's Roger Harrabin greeted that report as though it clinched the case for man-made global warming.
The 'conclusive paper' that AGW sceptics would be looking for would be a document that showed that climate models can predict the variation of average global temperature to a reasonable level of accuracy over at least a couple of decades in comparison with observed data. If that ever happens, climate science could then be regarded as a 'hard science' and scepticism of AGW would pretty much disappear.