What evidence or support is given for his "quasi-conversion"?
The term is used in a general summation of Constantine's life, but not elaborated on. Here is the source quote, in bold. I included the entire paragraph for context.
The Christianization of the Roman Empire is typically divided into two phases, before and after the year 312 which marked the momentous quasi-conversion of Constantine. By this date Christianity had already converted a significant but unknown proportion of at least the urban population of the Empire, including a number of the elite classes. Constantine ended the intermittent persecution of Christianity with the Edict of Milan, which granted tolerance to all religions, specifically mentioning Christianity. Under Constantine's successors, Christianization of Roman society proceeded by fits and starts, as John Curran recently documented in detail.
I couldn't find an answer on /r/AskHistorians via the search function, but if I missed it I apologize - a link would be greatly appreciated! Thanks!
I don't think there's any particularly strong evidence supporting the view that it was a 'quasi-conversion'. I outlined the circumstances for his conversion here. I didn't really take a stance on this issue in that answer, but currently I definitely lean towards the view that it was a genuine conversion. We can for instance trace how the public image of his beliefs changed over time and his later actions can only be seen as evidence for his strong belief in Christianity. There were no doubt many 'cynical' factors for his conversion, but that did not mean that the conversion itself was not genuine.
Could you list the exact source of the quote?