This forum is about wrong numbers in science, politics and the media. It respects good science and good English.
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.
Further to my previous post, I just noticed that the WUWT blog post from 2015 includes a link to the NASA webpage which provides the ozone hole area data, and this webpage now includes data for the current year 2016:
ozone hole data
Using the same dataset as adopted in the WUWT blog post graph, which relates to the mean ozone hole area measured over the specific period Sept 7th to Oct 13th (presumably the hole area is at its biggest over that particular period of time), the area for 2016 is 20.9 million km², a drop from the 2015 value, but is the same as it was in 2014.
The other hole area versus time dataset on the NASA webpage gives the maximum area of the hole and the date it occurred, and for 2016 the maximum area is 23.0 million km², which represents a drop from the 2015 value, but the 2016 area is still higher than the maximum area for the hole seen in several previous years of the 21st century, 2012, 2010, 2004 and 2002.
So I don't see any strong evidence that the ozone hole is recovering.
As mentioned in the WUWT blog post, the people who came up with the 'CFCs are destroying the ozone layer' theory (Paul J. Crutzen, Mario J. Molina and F. Sherwood Rowland) were awarded a Nobel Prize for Chemistry in 1995. I remember thinking it was a bit premature to award them the prize in 1995 - you'd have thought that the Nobel Prize Committee would at least have waited for the ozone hole to close up before giving them it.
Well I have to say that where I work now, a colleague of mine was in charge of the AEA laboratory that designed and built the instruments that measured the ozone layer for the ozone hole investigation project.
He told me that at the end of it, the research scientists had admitted to him that they had no idea if there had always been a hole in the ozone layer.
On the possibility of there always having been an ozone hole present in the Antarctic region, there was quite a big boost for the idea that ozone holes can occur naturally in the Earth's atmosphere when a small ozone hole suddenly opened up over the Arctic region in 2011:
Arctic ozone hole
I don't remember the issue of there being an Arctic ozone hole in the last few years being reported in the UK mainstream media at all. The existence of this new hole, which has occurred after the phase-out of CFC gases, does not fit in with the Green narratives that all UK environmental journalists and most UK science journalists tend to follow.
But an Arctic ozone hole is potentially more dangerous and should be more newsworthy than having one present over the Antarctic region. The Antarctic is an uninhabited continent, apart from a small number of scientists who work there on a temporary basis. If the Arctic ozone hole grew in size, it might eventually affect Canada and Northern Europe.