His explanation that they had not even been allowed by Glasgow 2014 organisers to distribute anything promoting the Scottish Open Grand Prix that will take place at the same venue later this year met scepticism.
"I really don't think he believed me and thought I was just passing the buck," said a frustrated Conway.
The reality is that the organisers even banned Cyril the Squirrel, Badminton Scotland's mascot, from attending the Games lest, God forbid, he queer Clyde's pitch.
That brings me back to the article that filled this space on the opening day of the Games, when I explained how those running them would produce a superb show but had no interest in "legacy" implications.
The only legacy they are really interested in is their own, in terms of their next job, and while that is understandable, it means that unless others are prepared, and allowed, to fill the void then the Scottish public will have been duped in terms of the main justification of the vast public expenditure.
I write this in full awareness that my views may be attacked as curmudgeonly at a time when we should be celebrating the great success that was the Commonwealth Games in Glasgow.
Let me be very clear, no-one took more pleasure out of seeing all these sports given a platform than me.
Throughout my career, but this past year in particular, it has been my personal and professional mission to try to broaden sporting focus beyond the damaging national obsession with football, its tribalism and, in Scotland, its sheer mediocrity. Much as I still love to play and watch it, football has contributed more than anything else to feeding the worst characteristics of Scots, not least the belief that we are no good at sport.
The final Games medal count told a different story, 19 golds and 15 silvers showing that, far from always finding ways of snatching defeat from the jaws of victory, those Scottish kids who gave themselves a real chance of winning did so more than 50% of the time.
The problem is not lack of "bottle", it is a lack of opportunity for enough of them to pursue things they are good at and to be properly rewarded for it.
We cannot, though, disengage our brains and offer a purely emotional response to what we saw over the past fortnight, as so many media outlets did during the Games. To that end it was a relief to discover that the BBC had rediscovered the capacity for a bit of objective journalism immediately afterwards with a report on the work of Professor Simon Shibli, director of the Sport Industry Research Centre at Sheffield Hallam University, which warned of the dangers of reading too much into this one event, however welcome it was.
Shibli had, ahead of the Games, apparently predicted with extraordinary accuracy how many medals Scotland would win and explained just why the number would be hugely boosted compared with the previous few Games, not least because of the scope given to host nations to choose which sports are included.
Remove the medals won in judo, para-sports and gymnastics - none of which were included in 1986 - and we are immediately well below the medal tally of 33 achieved that year.
Once more, then, it is valid to ask exactly what has been achieved by the legions of sports administrators employed in the interim?
Even if we accept the "record" tally at face value, the huge increase in the number of available medals must be taken into account and, as I promised a few weeks ago, I have updated "the real medal table", showing medals won as a proportion of those available.
Just as happened in India last time around, hosting the Games has prompted a doubling of the medal tally compared with the previous Games, but even then the evidence is that all the public money invested still leaves us worse off, in real terms, than at each of the previous two home Games.
No amount of smart accountancy should, then, be allowed to disguise the fact that it will have been mere indulgence of a few if it does not lead to better outcomes in terms of opportunities for our youngsters.
The real target, of course, is for sport to play its proper part in improving the health of a nation which has long been considered a cause for major concern.
It is vital that all those Scots who attended events are now given further opportunity to participate in sport and it will be a disgrace if, as has happened before, officialdom prevents that happening.
Those who genuinely care about the health of this nation must now be assisted in every way to get on with improving the situation rather than being blocked by careerists.