> This is a departure from the discussion of supplementing vitamin D though, isn't it?
I don't think so, no. If my hypothesis is correct, the problem is not that they didn't go outside and play sports. The problem is that they are sick, and sick people don't go outside as much.
Forget Vitamin D for a moment. Imagine that every time you were outside for an hour, the Outside Fairy granted you one Outside Token. Over the course of a month, it's reasonable to assume that healthy people would have more Outside Tokens than sick people. But these Outside Tokens obviously would have no causal effect on whether or not people are sick. Giving sick people Outside Tokens would obviously not help them - their problem is that they are sick. They have less Outside Tokens because being sick makes it more difficult to be outside for an hour.
In other words, the arrow of causation is being sick -> less Outside Tokens, and not being sick -> more Outside Tokens. If that's the case, then Outside Tokens are irrelevant to treating the sick people. If, say, they have cancer, you should ignore the Outside Tokens and treat the cancer with all of the standard means we have of treating cancer.
They hypothesis is that "low levels of Vitamin D" are just those Outside Tokens.
(Yes, when Vitamin D levels are low enough, that can cause other kinds of health problems. But this hypothesis is that what we're measuring as "low" is still within the healthy range. How do we know when it's not in the healthy range? When it tends to cause the kinds of problems we are more confident are directly linked to low levels of Vitamin D.)