In the years 774-775 and 993-994, there were large peaks in Earth's atmospheric radiocarbon (14C) concentration. While the cause was initially disputed, Mekhaldi et al.'s 2015 study of annually resolved 10Be measurements as well as lower-resolved 36Cl data from ice cores seems to have resolved the debate. They concluded that the cause was two extremely large solar proton events, with the 774-5 event described as "at least five times stronger than any instrumentally recorded solar event". A solar origin for these 14C 'excursions' seems to be well-accepted at this point - while I have not looked at every single paper that cites Mekhaldi et al., I have not yet come across any serious disagreements with its conclusions from any paper published after it came out, and have seen several that seem to treat it as the final word.
As I understand it, an extreme solar proton event could have been caused by either a very large CME (which would have caused a large geomagnetic storm) or a very large solar flare (which would not have, unless it was accompanied by a large CME). So, from the perspective of better estimating the probability that Earth will be hit with very large geomagnetic storms in the future, the cause of the events seems relevant.
First, is my understanding that either a large flare or a large CME could have caused a large solar proton event correct? Or perhaps it does not matter from the perspective of risk estimation, if there is theoretical or empirical evidence that indicates that it is extremely likely that an extremely large flare would be very likely to be associated with a very large CME (is there)? Finally, is R. A. Harrison's 1995 observation that "The characteristics of CME-associated flares (brightness, duration, etc.) appear to have no relationship to the characteristics of the associated CME (velocity, size, etc...)" still believed to be true - and do we have any inkling of whether it would also likely be true of extremely large flares? Feeling slightly over my head as a non-astronomer reading up on the topic, and would appreciate any pointers that could shed light on any of these questions.
p.s. Some potentially relevant reading I have done: It looks like there have been various attempts to quantify the number of flares associated with CMEs - for example, Harrison notes in an analysis of flares and CMEs during 1986-7 that if one was to assume that for every flare there is a CME and vice versa, visibility constraints would mean that we would expect to have observed CMEs associated with 300 of the 674 observed flares, but CMEs were associated with only 72 of them. However, the higher the flare class, the more likely it was that a CME would be associated with it. A 2016 study seems to report that of 42 X-class flares studied, CMEs were associated with 79%, but I'm not sure to what degree I should assume that the processes capable of producing a flare "at least five times stronger than any instrumentally recorded solar event" resemble the process that produce normal X-class flares.