I would think it would be 1/3 the diameter or even less, since the shot load starts travelling as a mass inside the shot cup. Once the shot starts to spread, it will quickly start acting like a bunch of individual pellets rather than a unified mass, but it can travel some distance before that happens.
You are probably correct. I was being my usual conservative engineer self. But I hadn't thought about the shot not starting to spread until the petals pull off the shot-cup/wad. Those generally travel about 15-20 yards, but much of that is after it has separated from the shot.
So, for the first few yards, the difference between a slug and shot is pretty negligible, and even for the next few after that, the shot column doesn't spread much.
Bottom line, I hope I never get shot by a shotgun at close range. And if I ever had to employ my serious social purposes shotgun for it's intended use, I hope there's somebody else to clean up the mess.