Look at figure 2 in the Lancet study. Seems to be making that claim indeed.
As for immune function, the phrasing isn't as precise as it could be but he's paraphrasing a cited study so does it have to be? There's lots of fraud in science but that word should be reserved for things that justify it, like made up data, misrepresenting your own data, motivated reasoning etc. Assuming the reader will understand what's meant based on context, especially when the cited study can be easily loaded, doesn't rise to the level of fraud.
As for immune function, the phrasing isn't as precise as it could be but he's paraphrasing a cited study so does it have to be? There's lots of fraud in science but that word should be reserved for things that justify it, like made up data, misrepresenting your own data, motivated reasoning etc. Assuming the reader will understand what's meant based on context, especially when the cited study can be easily loaded, doesn't rise to the level of fraud.