So, geography plays a part in it, it seems. Rates are higher in the PacNW, major issue here in Seattle, etc. Seasons seem to be a factor as well, on top of geography. Onset and relapses seem to occur most in the spring. You know, when the rest of the nation has sun, and winter is till dragging on and it's shitting rain sideways here in Seattle.
The researchers, or at least the summary seems aware that what if the actual correlation is not weight, but rather the indoor sedentary lifestyle, at least partially induced by the weather? This study didn't establish the weight as a cause, or even a factor. First best guess, I'd look to how much time these people spend outdoors, getting a little sun, getting fresh air and exercise, etc.
Like me, the weight is probably a factor because of an indoor lifestyle, and that is probably an indicator, rather than a cause, of associated MS risk.