She is Scotland's greatest all-round athlete, competing here in a record-equalling (with Liz McColgan) third Games. But as the curtain comes down on her Olympic career, with a likely 4 x 400m appearance, it's worth looking back.
She has won World Championship relay bronze, European 400m bronze and relay bronze, Commonwealth 400m silver and 400m hurdles bronze, and World Student Games relay silver. She was a finalist in the individual 400m there, and a high-jump competitor. She even reached the world individual 400m final in Paris, and won European under-23 relay bronze.
Anything missing? Yes, an Olympic medal. Now, she might win one by the end of this week, but that's not the point. She should already have one.
In 2004, in Athens, Britain were fourth in the 4 x 400m, behind the USA, Russia and Jamaica, with a time of 3:25.12. She ran the anchor leg in a squad which read Donna Fraser, Cath Murphy, Christine Ohuruogu, and McConnell.
Two years ago, Crystal Cox, who ran in the heats, accepted a four-year doping ban due to evidence which emerged in an FBI inquiry into the BALCO affair. She signed a confession. She had used illegal performance-enhancing drugs in the three years to 2004.
The IAAF confirmed to The Herald this week that it disqualified the US quartet (DeeDee Trotter, Monique Henderson, Sanya Richards, and Monique Hennagan) plus heat runner Cox, two years ago, and duly informed the International Olympic Committee.
They had disqualified Cox under a rule which they had introduced just months before the 2004 Games. It stipulated that if a relay runner is banned for a doping offence, the whole team is sanctioned, forfeiting their medals.
A fortnight past Saturday, the IOC said they had stripped Cox of her medal. So surely the Britons now get their bronze?
I put this to the International Association of Athletics Federations yesterday. They said the ball was in the IOC court, because the Games were under their jurisdiction, not that of the IAAF. Had it been an IAAF event, the medals would already be round the British women's necks.
I spoke to Sir Craig Reedie, the Scot newly elected to the IOC executive board, now the second most powerful figure in the Olympic movement. He did not have all the papers to hand. Understandably. He was at the triathlon, watching the Brownlee medal charge. He referred me to Mark Adams, the IOC press officer.
I suggested to Reedie that the most appropriate course would be for the five British women to be lined up in the Olympic arena in London, and their medals presented there. Eight years is more than belated, but preferable to a Jiffy bag couriered to their home, as has happened in the past. Or a silver medal presented at a road race in Battersea Park, the fate of the 10,000m runner-up Mike McLeod in 1985. London would be appropriate. At least it would be in an Olympic arena, with the bonus of being in front of their own people. "That will not happen," said Reedie. He said the delay was because of the possibility of further revelations and sanctions on the back of the Balco affair which implicated numerous athletes in several sports. The biggest icon to fall was Marion Jones, who had survived 167 drug tests, but was jailed for lying to the FBI.
Sir Craig was confdent the issue would be resolved. So would they get their medals eventually? "I very much hope so," he said, "but talk to Mark."
Adams was rushing to a press conference, but professed to be "very interested" in the case. He indicated there were complex legal issues. "We are waiting for formal confirmation from the IAAF." But he said he would be delighted to brief me at length later. Ten phone calls and three texts later, we are still waiting.
Back to the IAAF. They seemed bemused. It seems the issue is being batted back and forward between lawyers, arguing over technicalities. So while two of the world's most powerful sports organisations keep delaying, justice is denied. These bodies are supposed to protect competitors' interests. Instead, they allow the innocent to continue being penalised when it is somebody else who has cheated
The threat of litigation from US athletes is doubtless real. They could argue that there is no guarantee Cox doped in Athens, despite her written testimony.
But the IOC could have made a very positive statement by presenting the medals in London. They have missed the chance of a public relations coup. Medals reallocated in a packed Olympic arena, before a TV audience of billions, would have made a very telling comment on their view of drug cheats.
McConnell says she experiences conflicting emotions when she thinks about it. "Having lost the opportunity to stand on a podium to get it, there would probably be a lot of anger and upset," she said. "But it would be nice to complete the set."
When the IOC executive board finally make up their minds up, they should fly all those who have been wronged to Rio de Janiero, and make the awards at the next Olympics.
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